Explanation:
Okay, so I need to find the minimum value of (x + y)(y + z) given that x, y, z are positive real numbers and satisfy the condition xyz(x + y + z) = 1. Hmm, let's see.

First, I should probably try to understand the problem and see if I can relate the given condition to the expression we need to minimize. The expression (x + y)(y + z) can be expanded, maybe that will help? Let me try expanding it:

(x + y)(y + z) = xy + xz + y² + yz.

Hmm, not sure if that helps directly. Maybe I can think of it in terms of variables. Let me note that the given condition is xyz(x + y + z) = 1. So, maybe I need to use some inequalities here? Like AM-GM inequality? Since we have products and sums, that might be useful.

But before that, perhaps I can make substitutions to reduce the number of variables? Let me think. The expression we need to minimize is (x + y)(y + z). Let me denote some variables. Let me think about symmetry. Is there a way to make some substitution that can simplify things?

Alternatively, maybe we can fix some variables? For example, maybe set some variables equal? Let me try to assume that x = z? Because sometimes symmetry gives the extremum. If x = z, maybe that would simplify things?

Let me try that. Suppose x = z. Then, let me denote x = z = a, and y = b. Then, the condition becomes:

xyz(x + y + z) = a * b * a * (a + b + a) = a²b(2a + b) = 1.

So, a²b(2a + b) = 1.

And the expression to minimize is (x + y)(y + z) = (a + b)(b + a) = (a + b)².

Oh! Wait, if x = z = a and y = b, then (x + y)(y + z) = (a + b)(b + a) = (a + b)^2. So, we need to minimize (a + b)^2, given that a²b(2a + b) = 1.

So, maybe this substitution can help? Let me think. Let me denote S = a + b, and maybe express the condition in terms of S and something else? Or maybe express in terms of variables?

Alternatively, maybe I can use Lagrange multipliers here? But since I'm trying to find a minimum, maybe there is a smarter way.

Alternatively, maybe set t = b/a? Let me try. Let me set t = b/a, so that b = ta. Then, substituting into the condition:

a² * ta * (2a + ta) = a² * ta * a(2 + t) = a^3 * t(2 + t) = 1.

Therefore, a^3 = 1/(t(2 + t)), so a = [1/(t(2 + t))]^{1/3}.

Then, S = a + b = a + ta = a(1 + t) = [1/(t(2 + t))]^{1/3} * (1 + t).

Therefore, (a + b)^2 = [ (1 + t)^3 / (t(2 + t)) ]^{2/3}.

So, to minimize (a + b)^2, we need to minimize [ (1 + t)^3 / (t(2 + t)) ]^{2/3} which is equivalent to minimizing (1 + t)^3 / (t(2 + t)).

Therefore, if I can find the minimum of f(t) = (1 + t)^3 / [t(2 + t)] for t > 0, then take that value and raise it to the 2/3 power, that will be the minimum of (a + b)^2.

So, let me focus on minimizing f(t) = (1 + t)^3 / [t(2 + t)] for t > 0.

To find the minimum of f(t), take derivative and set to zero. Let me compute f'(t). Let me denote numerator as N(t) = (1 + t)^3 and denominator as D(t) = t(2 + t) = 2t + t^2. Therefore, f(t) = N(t)/D(t).

So, f'(t) = [N'(t) D(t) - N(t) D'(t)] / [D(t)]².

First compute N'(t) = 3(1 + t)^2.

D'(t) = 2 + 2t. Wait, D(t) is t(2 + t) = 2t + t², so derivative is 2 + 2t? Wait, no. Wait, derivative of t(2 + t) is derivative of 2t + t² which is 2 + 2t. Yes, correct.

Therefore, f'(t) = [3(1 + t)^2 * (2t + t²) - (1 + t)^3 * (2 + 2t)] / (2t + t²)^2.

Let me factor out common terms. Let me note that numerator is:

(1 + t)^2 [3(2t + t²) - (1 + t)(2 + 2t)].

Let me compute the bracket:

First term: 3(2t + t²) = 6t + 3t²

Second term: (1 + t)(2 + 2t) = 2(1 + t)(1 + t) = 2(1 + t)^2 = 2(1 + 2t + t²) = 2 + 4t + 2t²

Therefore, the bracket is:

[6t + 3t²] - [2 + 4t + 2t²] = 6t + 3t² - 2 - 4t - 2t² = (6t - 4t) + (3t² - 2t²) - 2 = 2t + t² - 2.

So, numerator is (1 + t)^2 (t² + 2t - 2).

Therefore, critical points when numerator is zero. Since (1 + t)^2 is always positive for t > 0, we set t² + 2t - 2 = 0. Solve for t:

t = [-2 ± sqrt(4 + 8)] / 2 = [-2 ± sqrt(12)] / 2 = [-2 ± 2*sqrt(3)] / 2 = -1 ± sqrt(3).

Since t > 0, we take the positive root: t = -1 + sqrt(3). Wait, sqrt(3) is approx 1.732, so -1 + 1.732 ≈ 0.732 > 0. So, critical point at t = sqrt(3) - 1.

So, this is the critical point. Now we need to check if it's a minimum.

Let me check the second derivative or test intervals. Let me pick t approaching 0 from the right: for t approaching 0+, the numerator of f(t) is (1 + t)^3 ~ 1, and denominator t(2 + t) ~ 2t, so f(t) ~ 1/(2t) which tends to infinity. Similarly, as t approaches infinity, f(t) ~ t^3 / (t * t) = t^3 / t² = t, which tends to infinity. Therefore, the function tends to infinity at both ends, so the critical point must be a minimum.

Therefore, the minimum occurs at t = sqrt(3) - 1.

Now compute f(t) at t = sqrt(3) - 1:

First, compute t = sqrt(3) - 1.

Let me compute 1 + t = 1 + sqrt(3) - 1 = sqrt(3).

Compute denominator t(2 + t):

First compute 2 + t = 2 + sqrt(3) - 1 = 1 + sqrt(3).

Therefore, denominator is t*(1 + sqrt(3)) = (sqrt(3) - 1)*(1 + sqrt(3)) = (sqrt(3))^2 - 1^2 = 3 - 1 = 2.

Therefore, f(t) = (sqrt(3))^3 / 2.

Wait, numerator is (1 + t)^3 = (sqrt(3))^3 = 3*sqrt(3).

So, f(t) = 3√3 / 2.

Therefore, the minimal value of (a + b)^2 is [f(t)]^{2/3} = [ (3√3)/2 ]^{2/3}?

Wait, hold on. Wait, earlier we had (a + b)^2 = [ (1 + t)^3 / (t(2 + t)) ]^{2/3} = [f(t)]^{2/3}?

Wait, no. Wait, actually, earlier I had:

(a + b)^2 = [ (1 + t)^3 / (t(2 + t)) ]^{2/3} ?

Wait, let me retrace. Earlier:

We had S = a + b = [1/(t(2 + t))]^{1/3} * (1 + t). Therefore, (a + b)^2 = [ (1 + t)^3 / (t(2 + t)) ]^{2/3} ?

Wait, let me check:

If S = [1/(t(2 + t))]^{1/3} * (1 + t), then S^2 = [ (1 + t)^2 / (t(2 + t)) ]^{1/3} ?

Wait, no:

Wait, S = [ (1 + t) / (t(2 + t))^{1/3} ] ?

Wait, original substitution was that a^3 = 1 / [ t(2 + t) ], so a = [1 / (t(2 + t))]^{1/3}. Then S = a(1 + t) = [1 / (t(2 + t))]^{1/3} * (1 + t). Therefore, S^2 = [ (1 + t)^2 / (t(2 + t)) ]^{1/3} ?

Wait, so S^2 = [ (1 + t)^2 / (t(2 + t)) ]^{1/3} ?

But earlier I thought that (a + b)^2 is [ (1 + t)^3 / (t(2 + t)) ]^{2/3}... Wait, maybe I made a miscalculation earlier.

Let me redo that part. Let me clarify.

Original expression for S = a + b.

We had:

a^3 = 1 / [ t(2 + t) ]

Therefore, a = [1 / (t(2 + t))]^{1/3}

Then S = a(1 + t) = [1 / (t(2 + t))]^{1/3} * (1 + t)

Therefore, S^2 = [ (1 + t)^2 / (t(2 + t)) ]^{1/3}

Wait, so S^2 is equal to [ (1 + t)^2 / (t(2 + t)) ]^{1/3}

But earlier I thought that it was [ (1 + t)^3 / (t(2 + t)) ]^{2/3}, but that seems incorrect.

Wait, so if f(t) = (1 + t)^3 / [t(2 + t)], then [f(t)]^{2/3} would be [ (1 + t)^3 / (t(2 + t)) ]^{2/3} = (1 + t)^2 / [t(2 + t)]^{1/3}, which is the same as S^2. Therefore, yes, S^2 = [f(t)]^{2/3}. Therefore, since we found that the minimal value of f(t) is 3√3 / 2, then the minimal value of S^2 is [3√3 / 2]^{2/3}?

Wait, let me compute that:

Let me write 3√3 as 3 * 3^{1/2} = 3^{3/2}, so 3√3 / 2 = 3^{3/2}/2. Therefore, [3^{3/2}/2]^{2/3} = (3^{3/2})^{2/3} / 2^{2/3} = 3^{(3/2 * 2/3)} / 2^{2/3} = 3^1 / 2^{2/3} = 3 / 2^{2/3}.

But 2^{2/3} is the cube root of 4, so 3 divided by cube root of 4. But is that the minimal value? Wait, but maybe there is a simpler form? Alternatively, maybe I made a mistake in calculation earlier?

Wait, let me check again:

We found that f(t) = (1 + t)^3 / [t(2 + t)] reaches its minimum at t = sqrt(3) - 1, and the minimal value is 3√3 / 2? Wait, earlier when I computed f(t) at t = sqrt(3) - 1, I had:

(1 + t)^3 / [t(2 + t)] = (sqrt(3))^3 / 2 = (3*sqrt(3))/2. Yes, that's correct. Therefore, f(t) is minimized at 3√3 / 2. Therefore, S^2 = [f(t)]^{2/3} = [ (3√3)/2 ]^{2/3} ?

Alternatively, maybe we can write this as something else. Let me compute [ (3√3)/2 ]^{2/3}:

First, note that 3√3 = 3^{3/2}, so (3^{3/2}/2)^{2/3} = 3^{(3/2)*(2/3)} / 2^{2/3} = 3^1 / 2^{2/3} = 3 / 2^{2/3}.

Alternatively, rationalizing, 2^{2/3} is cube root of 4, so it's 3 divided by cube root of 4. But is there a better way to express this?

Alternatively, maybe I made a wrong assumption by setting x = z? Maybe the minimum occurs when x ≠ z? I need to check if this assumption is valid.

Alternatively, maybe I should try using Lagrange multipliers for the original problem without assuming x = z.

Let me try that approach as a check. Let me define variables x, y, z > 0, with constraint xyz(x + y + z) = 1, and want to minimize (x + y)(y + z).

Let me denote the function to minimize as f(x, y, z) = (x + y)(y + z). The constraint is g(x, y, z) = xyz(x + y + z) - 1 = 0.

Using Lagrange multipliers, we set ∇f = λ∇g.

Compute gradients:

First, compute partial derivatives of f:

df/dx = derivative of (x + y)(y + z) with respect to x is (1)(y + z) = y + z.

df/dy = derivative with respect to y is (x + y)'*(y + z) + (x + y)*(y + z)' = (1)(y + z) + (x + y)(1) = y + z + x + y = x + 2y + z.

df/dz = derivative with respect to z is (x + y)*(1) = x + y.

Now compute partial derivatives of g:

First, expand g(x, y, z) = xyz(x + y + z) - 1. Let me write it as xyz(x + y + z) - 1.

Compute dg/dx: derivative of xyz(x + y + z) with respect to x.

Use product rule: derivative of xyz * (x + y + z) + xyz * derivative of (x + y + z).

Wait, actually, better to expand first:

xyz(x + y + z) = x^2 y z + x y^2 z + x y z^2.

So, dg/dx = 2xyz + y^2 z + y z^2.

Similarly, dg/dy = x^2 z + 2x y z + x z^2.

dg/dz = x^2 y + x y^2 + 2x y z.

Wait, let me check:

Alternatively, using product rule:

g = xyz(x + y + z). Let me denote u = xyz, v = x + y + z. Then dg/dx = du/dx * v + u * dv/dx.

du/dx = yz, dv/dx = 1. Therefore, dg/dx = yz(x + y + z) + xyz(1) = yz(x + y + z + x) = yz(2x + y + z). Wait, that seems different from expanding. Wait, maybe mistake here?

Wait, no:

Wait, if u = xyz, then du/dx = yz. Then dv/dx = derivative of (x + y + z) w.r. to x is 1. Therefore, dg/dx = yz*(x + y + z) + xyz*1 = yz(x + y + z) + xyz = yz(x + y + z + x) ??? Wait, no:

Wait, it's yz(x + y + z) + xyz. Let me factor yz:

yz(x + y + z + x) ? Wait, no. Let me compute:

yz(x + y + z) + xyz = yz(x + y + z + x) ? No, that would be adding yz*x + yz*y + yz*z + xyz. Wait, no. Let me compute:

Original terms:

yz(x + y + z) = yzx + yzy + yzz = xyz + y²z + yz²

Plus xyz from the other term: total is 2xyz + y²z + yz². Which matches the expansion. Therefore, dg/dx = 2xyz + y²z + yz². Similarly for others.

So, the gradient of g is:

dg/dx = 2xyz + y²z + yz²,

dg/dy = 2xyz + x²z + xz²,

dg/dz = 2xyz + x²y + xy².

Therefore, setting ∇f = λ∇g:

So,

y + z = λ(2xyz + y²z + yz²) ...(1)

x + 2y + z = λ(2xyz + x²z + xz²) ...(2)

x + y = λ(2xyz + x²y + xy²) ...(3)

Additionally, we have the constraint xyz(x + y + z) = 1 ...(4)

This seems complex. Maybe we can assume some symmetry here. For example, if we assume that x = z, as before, then maybe equations simplify?

Let me try that. Let me set x = z. Then, variables are x, y, x. Let me denote x = z.

Then, let me compute equations (1), (2), (3):

Equation (1): y + x = λ(2x y x + y²x + y x²) = λ(2x² y + x y² + x² y) = λ(3x² y + x y²).

Equation (2): x + 2y + x = λ(2x y x + x² x + x x²) = λ(2x² y + x³ + x³) = λ(2x² y + 2x³).

Equation (3): x + y = λ(2x y x + x² y + x y²) = λ(2x² y + x² y + x y²) = λ(3x² y + x y²).

Therefore, equations (1) and (3) are the same when x = z. Therefore, equations (1) and (3) are same, so we have:

From equation (1): y + x = λ(3x² y + x y²) = λ x y (3x + y).

From equation (2): 2x + 2y = λ(2x² y + 2x³) = 2λx² (y + x).

Therefore, let me denote equation (1): x + y = λ x y (3x + y).

Equation (2): 2(x + y) = 2λ x² (x + y).

Assuming x + y ≠ 0 (which it isn't, since x, y > 0), we can divide both sides by (x + y):

From equation (2): 2 = 2λ x² => λ = 1 / x².

From equation (1): x + y = (1 / x²) * x y (3x + y) => x + y = (y / x)(3x + y).

Multiply both sides by x:

x(x + y) = y(3x + y)

Expand left side: x² + x y

Right side: 3x y + y²

Bring all terms to left:

x² + x y - 3x y - y² = x² - 2x y - y² = 0.

So:

x² - 2x y - y² = 0

Let me write it as x² - 2x y - y² = 0.

This is quadratic in x: x² - 2y x - y² = 0.

Solving for x:

x = [2y ± sqrt(4y² + 4y²)] / 2 = [2y ± sqrt(8y²)] / 2 = [2y ± 2y√2]/2 = y(1 ± √2).

Since x > 0 and y > 0, we discard the negative solution, so x = y(1 + √2). Therefore, x = y(1 + √2).

So, with x = z = y(1 + √2). Let me denote this as x = z = k y, where k = 1 + √2.

Now, let's use the constraint equation (4): xyz(x + y + z) = 1.

Substituting x = z = k y:

(k y) * y * (k y) * (k y + y + k y) = k² y³ * ( (2k + 1) y ) = k² (2k + 1) y^4 = 1.

Therefore, y^4 = 1 / [k² (2k + 1)].

Compute k = 1 + √2 ≈ 2.414, but let me keep it exact.

First compute 2k + 1 = 2(1 + √2) + 1 = 2 + 2√2 + 1 = 3 + 2√2.

Therefore, y^4 = 1 / [k² (3 + 2√2)].

But k = 1 + √2. Let me compute k²:

k² = (1 + √2)^2 = 1 + 2√2 + 2 = 3 + 2√2.

Therefore, k² (3 + 2√2) = (3 + 2√2)^2.

Compute (3 + 2√2)^2 = 9 + 12√2 + 8 = 17 + 12√2.

Therefore, y^4 = 1 / (17 + 12√2).

But perhaps we can rationalize the denominator? Let me see:

Multiply numerator and denominator by (17 - 12√2):

Wait, but denominator is 17 + 12√2. Let me check:

Wait, but wait, y^4 = 1 / [ (3 + 2√2)^2 ] = 1 / [ (sqrt(3 + 2√2))^4 ]... Wait, but 3 + 2√2 is (1 + √2)^2, so (3 + 2√2)^2 = (1 + √2)^4. Therefore, y^4 = 1 / (1 + √2)^4, so y = [1 / (1 + √2)^4]^{1/4} = 1 / (1 + √2).

Therefore, y = 1 / (1 + √2). Rationalizing:

Multiply numerator and denominator by (√2 - 1):

y = (√2 - 1)/[(1 + √2)(√2 - 1)] = (√2 - 1)/(2 - 1) = √2 - 1.

So y = √2 - 1. Then, x = z = k y = (1 + √2)(√2 - 1) = (√2)^2 - 1^2 = 2 - 1 = 1.

So x = z = 1, y = √2 - 1.

Therefore, let me check the value of (x + y)(y + z). Since x = z = 1, and y = √2 - 1.

Compute (x + y) = 1 + √2 - 1 = √2.

Similarly, (y + z) = √2 - 1 + 1 = √2. Therefore, (x + y)(y + z) = √2 * √2 = 2.

Wait, so the minimal value is 2? But earlier when I did the substitution with x = z, I found that the minimal value was 3 / 2^{2/3} ≈ 3 / 1.587 ≈ 1.89, which is less than 2. But that contradicts this result? Which one is correct?

Wait, there must be a mistake in my previous reasoning. Let me check.

Earlier, when I set x = z, then found that (a + b)^2 is [f(t)]^{2/3} where f(t) had a minimum of 3√3 / 2. Then, [ (3√3)/2 ]^{2/3}... Let me compute that numerically.

Let me compute 3√3: √3 ≈ 1.732, so 3*1.732 ≈ 5.196. Divided by 2: ≈ 2.598. Then take 2/3 power: 2.598^(2/3). Let me compute ln(2.598) ≈ 0.955. Multiply by 2/3: ≈ 0.637. Exponentiate: e^0.637 ≈ 1.89. But according to the Lagrange multipliers approach, the minimal value is 2. Which is higher. Therefore, there must be an error in one of the approaches.

But since Lagrange multipliers gave us an exact value of 2, and the substitution approach gave us approx 1.89, which is lower, but maybe I made a mistake in the substitution approach?

Wait, but when using the Lagrange multipliers with x = z, we found that the minimal value is 2. But if there exists another case where x ≠ z which gives a lower value, then that approach would be wrong. However, since Lagrange multipliers should account for all possibilities, perhaps there was a miscalculation in substitution approach?

Let me check substitution approach again.

Earlier, when I set x = z = a, y = b, then S = a + b. The expression to minimize was S^2 = [ (1 + t)^2 / (t(2 + t)) ]^{1/3} where t = b/a. Then I tried to minimize f(t) = (1 + t)^3 / [ t(2 + t) ] and found its minimum at t = sqrt(3) - 1, leading to S^2 = [3√3 / 2]^{2/3} ≈ 1.89. However, according to the Lagrange multipliers, when x = z, the minimum is achieved at 2. Therefore, these two results are conflicting. Which is correct?

Wait, let me check with actual numbers. If in the Lagrange approach, with x = z = 1, y = √2 - 1 ≈ 0.414, then let's compute the original condition xyz(x + y + z) = 1 * (√2 - 1) * 1 * (1 + (√2 - 1) + 1) = (√2 - 1) * (1 + √2 - 1 + 1) = (√2 - 1)(1 + √2) = (√2 - 1)(√2 + 1) = 2 - 1 = 1. Correct. So that satisfies the condition. And (x + y)(y + z) = 2.

Now, let me check with the substitution approach. If I take t = sqrt(3) - 1 ≈ 0.732. Then, let me compute b = t*a, and from earlier, a^3 = 1 / [ t(2 + t) ].

Let me compute t(2 + t): t ≈ 0.732, 2 + t ≈ 2.732. Multiply: 0.732 * 2.732 ≈ 2. So, a^3 ≈ 1/2, so a ≈ 0.7937. Then, b = t*a ≈ 0.732 * 0.7937 ≈ 0.581. Then, (x + y)(y + z) = (a + b)^2 ≈ (0.7937 + 0.581)^2 ≈ (1.3747)^2 ≈ 1.89, which is indeed less than 2. So according to this, there's a lower value. But why does the Lagrange multipliers method give a higher value?

Ah! Because in the Lagrange multipliers approach, when I assumed x = z, I found a critical point, but perhaps there is another critical point when x ≠ z, which gives a lower value. However, in the Lagrange multipliers approach, we set up equations and found that assuming x = z leads to a critical point, but maybe there are other critical points with x ≠ z. However, since the substitution approach with x = z led to a lower value, but the Lagrange multipliers didn't consider that? Or maybe I made a mistake in the substitution approach?

Wait, no. The substitution approach with x = z is a special case, so the minimal value could be either in that case or in another case. Therefore, the Lagrange multipliers method, when assuming x = z, gives a critical point with value 2, but there's another critical point (with x ≠ z) that gives a lower value. However, perhaps my mistake is in the substitution approach. Let me check again.

Wait, in the substitution approach, we set x = z and found that the minimal value of (x + y)(y + z) was approximately 1.89, but when I calculated with Lagrange multipliers with x = z, I get 2. So there is inconsistency. Therefore, I must have made a mistake in one of the steps.

Let me check the substitution approach again.

Earlier, I had:

After setting x = z = a, y = b, then the constraint is a²b(2a + b) = 1. Then, we set t = b/a, so b = t a. Therefore, the constraint becomes a² * t a * (2a + t a) = a³ t (2 + t) = 1 => a³ = 1/(t(2 + t)) => a = [1/(t(2 + t))]^{1/3}.

Then, (x + y)(y + z) = (a + b)^2 = [a(1 + t)]². So [a(1 + t)]² = [ (1 + t)^2 / (t(2 + t)) ]^{2/3} ?

Wait, hold on. If a = [1/(t(2 + t))]^{1/3}, then [a(1 + t)]² is [ (1 + t)^2 / (t(2 + t))^{1/3} ] ?

Wait, no. Wait:

a(1 + t) = [1/(t(2 + t))]^{1/3} * (1 + t)

Therefore, [a(1 + t)]² = [ (1 + t)^2 / (t(2 + t))^{1/3} ] ?

Wait, no. Let me compute:

[a(1 + t)]^2 = [ (1 + t) / (t(2 + t))^{1/3} ]² = (1 + t)^2 / [ t(2 + t) ]^{2/3}

But earlier I thought it was [ (1 + t)^3 / (t(2 + t)) ]^{2/3}, which is different. So I had an error in my substitution approach.

So the correct expression is (a + b)^2 = (1 + t)^2 / [ t(2 + t) ]^{2/3}

Therefore, to minimize this expression, we need to minimize f(t) = (1 + t)^2 / [ t(2 + t) ]^{2/3}

But previously, I incorrectly thought I needed to minimize (1 + t)^3 / [ t(2 + t) ], which led to a different result. So that was my mistake! Therefore, let me correct this.

So, to minimize f(t) = (1 + t)^2 / [ t(2 + t) ]^{2/3} over t > 0.

Let me denote this as f(t) = [(1 + t)^2] / [t(2 + t)]^{2/3}. To minimize this, take natural logarithm to make differentiation easier.

Let me set ln f(t) = 2 ln(1 + t) - (2/3)(ln t + ln(2 + t)).

Take derivative with respect to t:

d/dt [ln f(t)] = 2/(1 + t) - (2/3)[1/t + 1/(2 + t)].

Set derivative to zero:

2/(1 + t) - (2/3)[1/t + 1/(2 + t)] = 0.

Multiply both sides by 3 to eliminate denominators:

6/(1 + t) - 2[1/t + 1/(2 + t)] = 0.

Simplify:

6/(1 + t) = 2[1/t + 1/(2 + t)].

Divide both sides by 2:

3/(1 + t) = 1/t + 1/(2 + t).

Let me compute RHS: 1/t + 1/(2 + t) = [ (2 + t) + t ] / [ t(2 + t) ] = (2 + 2t)/[t(2 + t)] = 2(1 + t)/[t(2 + t)].

Therefore:

3/(1 + t) = 2(1 + t)/[t(2 + t)]

Multiply both sides by (1 + t):

3 = 2(1 + t)^2 / [t(2 + t)]

Multiply both sides by t(2 + t):

3t(2 + t) = 2(1 + t)^2.

Expand both sides:

Left side: 6t + 3t².

Right side: 2(1 + 2t + t²) = 2 + 4t + 2t².

Bring all terms to left:

6t + 3t² - 2 - 4t - 2t² = 0 => (6t - 4t) + (3t² - 2t²) - 2 = 2t + t² - 2 = 0.

So, t² + 2t - 2 = 0, which is the same equation as before! Therefore, t = [ -2 ± sqrt(4 + 8) ] / 2 = [ -2 ± sqrt(12) ] / 2 = -1 ± sqrt(3). As before, t = sqrt(3) - 1 ≈ 0.732.

Therefore, even after correcting my substitution approach, the critical point is same t = sqrt(3) - 1. Therefore, compute f(t) at this t:

f(t) = (1 + t)^2 / [ t(2 + t) ]^{2/3}.

First compute 1 + t = sqrt(3) as before. Then, t(2 + t) = (sqrt(3) - 1)(1 + sqrt(3)) = 3 - 1 = 2 as before. So [ t(2 + t) ] = 2. Therefore, f(t) = (sqrt(3))² / [2]^{2/3} = 3 / 2^{2/3} ≈ 3 / 1.587 ≈ 1.89.

But in the Lagrange multipliers approach, with x = z, I found that the minimal value is 2. So which one is correct?

Wait, let me check with actual numbers. Let me take t = sqrt(3) - 1 ≈ 0.732. Then, a³ = 1 / [t(2 + t)] = 1/2, so a ≈ (1/2)^{1/3} ≈ 0.7937. Then, b = t * a ≈ 0.732 * 0.7937 ≈ 0.581. Then, check if the original condition xyz(x + y + z) = 1:

x = a ≈ 0.7937, y ≈ 0.581, z = a ≈ 0.7937. Then, x + y + z ≈ 0.7937 + 0.581 + 0.7937 ≈ 2.168. Then, xyz(x + y + z) ≈ 0.7937 * 0.581 * 0.7937 * 2.168. Let me compute step by step:

First compute 0.7937 * 0.7937 ≈ 0.63, then multiply by 0.581: ≈ 0.63 * 0.581 ≈ 0.366. Then multiply by 2.168: ≈ 0.366 * 2.168 ≈ 0.793. Which is not 1. Wait, that's not equal to 1. That suggests that my substitution approach has an error?

Wait, no. Earlier, when I set x = z = a, y = b, then the constraint is a²b(2a + b) = 1. With t = b/a = sqrt(3) - 1, and a³ = 1/[t(2 + t)] = 1/2. Therefore, a = 2^{-1/3}. Then b = t * a = (sqrt(3) - 1) * 2^{-1/3}. Then, let me compute a²b(2a + b):

Compute a²b(2a + b) = a² * b * 2a + a² * b * b = 2a³b + a²b².

But given that a³ = 1/[t(2 + t)] = 1/2, and b = t a:

So 2a³b = 2*(1/2)*b = b. And a²b² = (a b)^2. But a b = a * t a = t a². Since a³ = 1/2, then a² = (1/2)^{2/3}, so a b = t * (1/2)^{2/3}.

Therefore, total expression: b + (t * (1/2)^{2/3})². But I need to check if this is equal to 1?

Wait, but according to the constraint, a²b(2a + b) = 1. Let me compute:

a²b(2a + b) = a²b*2a + a²b*b = 2a³b + a²b².

But since a³ = 1/(t(2 + t)) = 1/2, so 2a³b = 2*(1/2)*b = b.

Meanwhile, a²b² = (a b)^2 = [ a * b ]^2. But a b = a * t a = t a². Since a³ = 1/2, so a² = a^{-1} * a³ = a^{-1} * 1/2. Wait, perhaps better to compute numerically:

Given a³ = 1/2, so a = 2^{-1/3}, then a² = 2^{-2/3}, and b = t * 2^{-1/3}. Then, a²b² = [2^{-2/3} * t^2 * 2^{-2/3}] = t² * 2^{-4/3}.

Therefore, total expression: b + a²b² = t * 2^{-1/3} + t² * 2^{-4/3}. Let me plug t = sqrt(3) - 1 ≈ 0.732:

First term: 0.732 * 2^{-1/3} ≈ 0.732 * 0.7937 ≈ 0.581.

Second term: 0.732² * 2^{-4/3} ≈ 0.536 * 0.396 ≈ 0.212.

Total ≈ 0.581 + 0.212 ≈ 0.793, which is not 1. Wait, that contradicts the condition. Therefore, where is the mistake?

Wait, earlier I thought that with a³ = 1/[t(2 + t)] which is 1/2, so a²b(2a + b) = 1? But according to this, it is not. Therefore, my mistake was in the substitution approach. Where?

Originally, I said that with x = z = a, y = b, then the constraint is a²b(2a + b) = 1. Is this correct?

Yes, because xyz(x + y + z) = a * b * a * (a + b + a) = a²b(2a + b). So that is correct. Then, when I set t = b/a, then b = ta, and the constraint becomes a²*(ta)*(2a + ta) = a² * ta * a(2 + t) = a³ t(2 + t) = 1 => a³ = 1/[t(2 + t)]. Therefore, this is correct. Therefore, if we take t = sqrt(3) - 1, then a³ = 1/[t(2 + t)] = 1/2. Then, why does a²b(2a + b) not equal to 1? Let me compute it properly:

Compute a²b(2a + b):

= a²b*(2a + b)

But since a³ = 1/2 => 2a³ = 1 => 2a³ = 1. Also, since b = t a, then:

Let me compute a²b*(2a + b) = a² * t a * (2a + t a) = a³ * t (2 + t) = (1/2) * t(2 + t) = (1/2)*2 = 1.

Wait! Because t(2 + t) = 2? Earlier, when t = sqrt(3) - 1, we had t(2 + t) = 2. Yes! Therefore, a³ * t(2 + t) = (1/2)*2 = 1. So it does satisfy the constraint. Therefore, my previous calculation where I computed numerically was wrong. Let me check again with numbers.

Let me compute a³ = 1/2, so a ≈ 0.7937. Then, t = sqrt(3) - 1 ≈ 0.732. b = a*t ≈ 0.732 * 0.7937 ≈ 0.581. Then, x + y + z = a + b + a = 2a + b ≈ 2*0.7937 + 0.581 ≈ 1.5874 + 0.581 ≈ 2.1684. Then, xyz(x + y + z) = a * b * a * (x + y + z) = a²b*(x + y + z) ≈ (0.7937² * 0.581) * 2.1684 ≈ (0.63 * 0.581) * 2.1684 ≈ 0.366 * 2.1684 ≈ 0.793. Wait, but according to algebra, it should be 1. Why discrepancy?

Ah! Because I miscalculated the value of a. If a³ = 1/2, then a = (1/2)^{1/3} ≈ 0.7937, but let me use exact values. Let me calculate more accurately.

First, compute a³ = 1/2, so a = 2^{-1/3} ≈ 0.7937005259840998.

Then, t = sqrt(3) - 1 ≈ 1.7320508075688772 - 1 = 0.7320508075688772.

Then, b = a * t ≈ 0.7937005259840998 * 0.7320508075688772 ≈ let me compute:

First multiply 0.7937 * 0.73205:

Approximate:

0.7937 * 0.7 = 0.5556

0.7937 * 0.03205 ≈ 0.7937 * 0.03 ≈ 0.0238 + 0.7937 * 0.00205 ≈ 0.0238 + 0.001627 ≈ 0.0254

Total approx. 0.5556 + 0.0254 ≈ 0.581. So b ≈ 0.581.

Then, x + y + z = 2a + b ≈ 2*0.7937 + 0.581 ≈ 1.5874 + 0.581 ≈ 2.1684.

Now, compute xyz(x + y + z) = a² * b * (x + y + z).

Let me compute a² * b: a² = (2^{-1/3})² = 2^{-2/3} ≈ 0.63. Then, a² * b ≈ 0.63 * 0.581 ≈ 0.366. Then multiply by x + y + z ≈ 2.1684: 0.366 * 2.1684 ≈ 0.793. But according to algebra, it should be 1. Where is this discrepancy?

Wait, let me calculate exactly:

Since a³ = 1/2 and t(2 + t) = 2, then a³ * t(2 + t) = 1/2 * 2 = 1. Therefore, xyz(x + y + z) = 1. But why when I compute numerically it's about 0.793? Because of rounding errors in my approximate calculations. Let me try with more precise values.

Let me use exact expressions:

Let me compute a = 2^{-1/3}, b = t * a = (sqrt(3) - 1) * 2^{-1/3}.

Then, x + y + z = 2a + b = 2 * 2^{-1/3} + (sqrt(3) - 1) * 2^{-1/3} = [2 + sqrt(3) - 1] * 2^{-1/3} = [1 + sqrt(3)] * 2^{-1/3}.

Now compute xyz(x + y + z) = a² * b * (x + y + z) = a² * b * (x + y + z).

Compute step by step:

First, a² = 2^{-2/3}.

b = t * a = (sqrt(3) - 1) * 2^{-1/3}.

(x + y + z) = [1 + sqrt(3)] * 2^{-1/3}.

Multiply all: 2^{-2/3} * (sqrt(3) - 1) * 2^{-1/3} * [1 + sqrt(3)] * 2^{-1/3} =

Combine exponents of 2: -2/3 -1/3 -1/3 = -4/3.

Multiply the rest: (sqrt(3) - 1)(sqrt(3) + 1) = (sqrt(3))^2 - 1^2 = 3 - 1 = 2.

Therefore, total expression: 2 * 2^{-4/3} = 2^{1 - 4/3} = 2^{-1/3} ≈ 0.7937, which is not 1. Wait, this contradicts the earlier algebraic conclusion that it should be 1.

Wait, this is a problem. Earlier I thought a³ * t(2 + t) = 1, but it seems like it's not. Let me re-express:

Original constraint: a³ * t(2 + t) = 1.

But t(2 + t) = 2, so a³ * 2 = 1 => a³ = 1/2, which is correct. Therefore, xyz(x + y + z) = a²b(2a + b) = a²b(2a + b) = a³t(2 + t) = (1/2)*2 = 1.

But according to the calculation above, it's 2 * 2^{-4/3} = 2^{-1/3} ≈ 0.7937? What is wrong here?

Wait, no! Let me check again:

Wait, xyz(x + y + z) = a²b(2a + b). And we have:

a²b(2a + b) = a²b*2a + a²b*b = 2a³b + a²b².

But earlier, I thought this equals a³ * t(2 + t). But why?

Because I had: a²b(2a + b) = a² * b * (2a + b) = a³ * b * (2 + b/a) = a³ * b * (2 + t). But since b = t a, then yes, this is a³ * t a * (2 + t) = a^4 t (2 + t). Wait, this contradicts the earlier assertion. Wait, no, let me re-express:

Wait, I think I have a mistake here. Let me redo:

Original expression: a²b(2a + b) = a²b * 2a + a²b * b = 2a³b + a²b².

But since b = t a, substitute:

= 2a³ * t a + a² * (t a)^2 = 2 t a^4 + t² a^4 = a^4(2t + t²).

But earlier I thought this was a³ * t(2 + t). Where is the error?

Ah, no! Previously, I thought that a²b(2a + b) = a³ * t(2 + t). But that is incorrect. It should be a^4(2t + t²). Therefore, my previous assertion that a³ * t(2 + t) = 1 was wrong. It should be a^4(2t + t²) = 1. Therefore, my mistake was here. Earlier, I thought the constraint was a³ * t(2 + t) = 1, but actually it's a^4(2t + t²) = 1. Wait, where did that come from?

Let me go back. Original constraint: xyz(x + y + z) = 1. With x = a, y = b, z = a. Then:

xyz(x + y + z) = a * b * a * (a + b + a) = a²b(2a + b) = 1.

With b = t a, so this becomes a² * t a * (2a + t a) = t a³ * a(2 + t) = t a^4 (2 + t) = 1.

Wait, so the constraint is t a^4 (2 + t) = 1, not a³ * t(2 + t) = 1. Earlier I had written a³ = 1/[t(2 + t)], which was wrong. That was the mistake!

Therefore, correcting that:

From the constraint: t a^4 (2 + t) = 1 => a^4 = 1 / [t(2 + t)] => a = [1 / (t(2 + t))]^{1/4}

Therefore, previous substitution approach had a mistake in the exponent of a.

Therefore, S = a + b = a + t a = a(1 + t) = [1 / (t(2 + t))]^{1/4} * (1 + t)

Thus, (x + y)(y + z) = (a + b)^2 = [ (1 + t)^2 / (t(2 + t)) ]^{1/2}

Wait, raising to power 1/2?

Yes, because S = [1 / (t(2 + t))]^{1/4} * (1 + t), so squaring it gives [ (1 + t)^2 / (t(2 + t)) ]^{1/2}

So, (x + y)(y + z) = sqrt( (1 + t)^2 / [ t(2 + t) ]) = (1 + t) / sqrt(t(2 + t))

Therefore, now, the expression to minimize is f(t) = (1 + t) / sqrt(t(2 + t)) for t > 0.

Previously, I had different exponents due to miscalculations. Now, this is the correct expression.

So, now, let's minimize f(t) = (1 + t)/sqrt(t(2 + t)). Let me square it to make it easier, since sqrt is increasing. Let me define g(t) = [f(t)]² = (1 + t)^2 / [t(2 + t)].

Wait, this is the same function f(t) that I incorrectly minimized earlier, but now with correct understanding, it's the same function as before. Wait, earlier when I corrected the substitution approach, I had this expression and found the critical point at t = sqrt(3) - 1.

So, let me re-express: the expression to minimize is g(t) = (1 + t)^2 / [t(2 + t)]. Earlier, I had found that its minimum is 3√3 / 2 at t = sqrt(3) - 1. Therefore, [f(t)]² = g(t), so the minimum of f(t) is sqrt(3√3 / 2). Wait, no. Wait, if g(t) has minimum value 3√3 / 2, then f(t) = sqrt(g(t)) has minimum value sqrt(3√3 / 2).

Wait, let me compute this value:

First, 3√3 / 2 is approximately 3 * 1.732 / 2 ≈ 5.196 / 2 ≈ 2.598. Then sqrt(2.598) ≈ 1.612.

But earlier, in the Lagrange multipliers approach, with x = z, I found that the minimal value was 2. Now, with corrected substitution approach, I find that it's approx 1.612, which is less than 2. So, this suggests that there is a lower value.

But let's check with exact values. At t = sqrt(3) - 1, we have:

g(t) = (1 + t)^2 / [t(2 + t)] = (sqrt(3))^2 / 2 = 3 / 2.

Wait, earlier I thought it was 3√3 / 2, but no. Wait, let me compute:

If t = sqrt(3) - 1, then 1 + t = sqrt(3), and t(2 + t) = 2. Therefore, g(t) = (sqrt(3))² / 2 = 3 / 2. Therefore, f(t) = sqrt(g(t)) = sqrt(3/2) ≈ 1.2247.

Wait, this is different from my previous calculation. What is going on?

Wait, previously, I thought that g(t) = (1 + t)^2 / [t(2 + t)] had a minimum of 3√3 / 2, but no, with t = sqrt(3) - 1, g(t) = 3 / 2. Wait, how did I get 3√3 / 2 before? That was a mistake. Let me retrace.

Earlier, when I was thinking of f(t) = (1 + t)^3 / [t(2 + t)], then at t = sqrt(3) - 1, it was (sqrt(3))^3 / 2 = 3√3 / 2. But that was for a different function. Now, with the corrected expression, g(t) = (1 + t)^2 / [t(2 + t)] has minimum value 3/2 at t = sqrt(3) - 1.

Therefore, the expression (x + y)(y + z) is sqrt(g(t)) = sqrt(3/2) ≈ 1.2247? Wait, no. Wait, the expression to minimize was f(t) = (1 + t)/sqrt(t(2 + t)) = sqrt(g(t)), and g(t) has minimum value 3/2, so f(t) has minimum value sqrt(3/2). Wait, sqrt(3/2) ≈ 1.2247.

But earlier, with the Lagrange multipliers approach, assuming x = z, we found that the minimal value is 2. How to resolve this?

Let me check with t = sqrt(3) - 1 ≈ 0.732, then compute (x + y)(y + z) = f(t) = (1 + t)/sqrt(t(2 + t)) = sqrt(3)/sqrt(2) = sqrt(3/2) ≈ 1.2247.

Let me verify this with the actual values.

With t = sqrt(3) - 1, and a^4 = 1 / [t(2 + t)] = 1/2, so a = (1/2)^{1/4} = 2^{-1/4} ≈ 0.8409. Then, b = t * a ≈ 0.732 * 0.8409 ≈ 0.616. Then, x + y + z = 2a + b ≈ 2*0.8409 + 0.616 ≈ 1.6818 + 0.616 ≈ 2.2978.

Now, compute xyz(x + y + z) = a² * b * (x + y + z) ≈ (0.8409²) * 0.616 * 2.2978. Compute step by step:

0.8409² ≈ 0.707.

Multiply by 0.616: 0.707 * 0.616 ≈ 0.435.

Multiply by 2.2978: 0.435 * 2.2978 ≈ 1.000. Yes! It satisfies the constraint.

Now, compute (x + y)(y + z). Since x = a ≈ 0.8409, y ≈ 0.616, z = a ≈ 0.8409. Then, (x + y)(y + z) = (a + y)(y + a) = (a + y)^2 ≈ (0.8409 + 0.616)^2 ≈ (1.4569)^2 ≈ 2.123. Wait, but sqrt(3/2) ≈ 1.2247, which is different. What is happening here?

Wait, no! Wait, my expression for (x + y)(y + z) is (a + b)^2, where b = y. But in reality, with x = a, y = b, z = a, (x + y)(y + z) = (a + b)(b + a) = (a + b)^2. But according to the previous substitution approach, this equals f(t) = sqrt(g(t)) = sqrt(3/2). But calculations show approximately 2.123. Which is different from sqrt(3/2) ≈ 1.2247.

This suggests another error in substitution approach. Where is the mistake?

Let me re-express: we had (x + y)(y + z) = (a + b)^2 = [a(1 + t)]^2 = a²(1 + t)^2.

But with a^4 = 1 / [t(2 + t)] => a² = 1 / [t(2 + t)]^{1/2}.

Therefore, (x + y)(y + z) = [1 / sqrt(t(2 + t))] * (1 + t)^2 = (1 + t)^2 / sqrt(t(2 + t)) = f(t).

But earlier, I wrote that this is equal to sqrt(g(t)), where g(t) = (1 + t)^2 / [t(2 + t)]. But actually, f(t) = g(t)^{1/2} * something? Wait, no:

Wait, g(t) = (1 + t)^2 / [t(2 + t)].

So f(t) = (1 + t)^2 / sqrt(t(2 + t)) = g(t) * sqrt(t(2 + t)) / sqrt(t(2 + t)) ? Wait, no:

Wait, f(t) = (1 + t)^2 / [t(2 + t)]^{1/2} = g(t)^{1/2} * [t(2 + t)]^{1/2} ? No, let me see:

Actually, g(t) is (1 + t)^2 / [t(2 + t)], and f(t) is (1 + t)^2 / sqrt(t(2 + t)) = g(t) * sqrt(t(2 + t)).

Wait, no, it's f(t) = g(t) * sqrt(t(2 + t))? Let me see:

Wait, g(t) = (1 + t)^2 / [t(2 + t)].

If I multiply g(t) by sqrt(t(2 + t)), I get [(1 + t)^2 / [t(2 + t)] ] * sqrt(t(2 + t)) = (1 + t)^2 / sqrt(t(2 + t)) = f(t).

So f(t) = g(t) * sqrt(t(2 + t)).

But earlier, when I computed g(t) at t = sqrt(3) - 1, it was 3/2. And sqrt(t(2 + t)) = sqrt(2). Therefore, f(t) = (3/2) * sqrt(2) ≈ 1.5 * 1.414 ≈ 2.121, which matches the numerical calculation of ≈ 2.123.

Ah! So my previous mistake was in thinking that f(t) was sqrt(g(t)), but it is actually g(t) * sqrt(t(2 + t)). Therefore, the expression to minimize is f(t) = g(t) * sqrt(t(2 + t)) = [(1 + t)^2 / [t(2 + t)]] * sqrt(t(2 + t)) = (1 + t)^2 / [t(2 + t)]^{1/2}.

Therefore, to find its minimum, we need to minimize this expression. However, earlier, I had found the critical point of g(t), which is (1 + t)^2 / [t(2 + t)], and found that its minimum is 3/2 at t = sqrt(3) - 1. But since f(t) = g(t) * sqrt(t(2 + t)), we need to minimize this product.

Wait, but how does this relate to the previous derivative calculation? Earlier, when I set up the derivative for g(t), I found that its minimum occurs at t = sqrt(3) - 1. Let me see what f(t) is at that point:

f(t) = g(t) * sqrt(t(2 + t)) = (3/2) * sqrt(2) ≈ 2.121, which matches the numerical calculation. So the minimum of f(t) is (3/2)*sqrt(2) = 3√2 / 2 ≈ 2.121. However, in the Lagrange multipliers approach with x = z, I found the value of (x + y)(y + z) as 2, which is less than 2.121. This is contradictory.

Wait, but earlier when I used the Lagrange multipliers with x = z, I found that the minimal value is 2, but according to this substitution approach, it's approximately 2.121. So which one is correct?

Let me check with the exact values from the Lagrange multipliers approach. When x = z = 1, y = sqrt(2) - 1 ≈ 0.414, then (x + y)(y + z) = (1 + y)^2. Let me compute it:

1 + y = 1 + sqrt(2) - 1 = sqrt(2), so (sqrt(2))^2 = 2.

But according to the substitution approach, when t = sqrt(3) - 1 ≈ 0.732, the value is approximately 2.121. Which is higher than 2. So this suggests that the minimum is 2, achieved at x = z = 1, y = sqrt(2) - 1.

But why does the substitution approach suggest a lower value? Because I think I made an error in interpreting the critical point. Earlier, I was trying to minimize g(t) = (1 + t)^2 / [t(2 + t)], but actually, the expression to minimize is f(t) = (1 + t)^2 / sqrt(t(2 + t)) = g(t) * sqrt(t(2 + t)).

So, to find the minimum of f(t), we cannot just minimize g(t), but need to consider the whole expression. Therefore, let me go back to the derivative.

Let me define f(t) = (1 + t)^2 / sqrt(t(2 + t)) = (1 + t)^2 / [t(2 + t)]^{1/2}. Let me take the derivative of f(t) with respect to t and set it to zero.

Let me set f(t) = (1 + t)^2 * [t(2 + t)]^{-1/2}.

Take logarithm: ln f(t) = 2 ln(1 + t) - (1/2) ln(t) - (1/2) ln(2 + t).

Take derivative:

f'(t)/f(t) = 2/(1 + t) - (1/2)(1/t) - (1/2)(1/(2 + t)).

Set to zero:

2/(1 + t) - 1/(2t) - 1/(2(2 + t)) = 0.

Multiply all terms by 2t(2 + t)(1 + t) to eliminate denominators:

2 * 2t(2 + t) - (1 + t)(2 + t) - (1 + t)t = 0.

Simplify term by term:

First term: 4t(2 + t) = 8t + 4t².

Second term: -(1 + t)(2 + t) = -(2 + t + 2t + t²) = - (2 + 3t + t²).

Third term: -(1 + t)t = -t - t².

Combine all terms:

8t + 4t² - 2 - 3t - t² - t - t² = 0.

Simplify:

Combine like terms:

t² terms: 4t² - t² - t² = 2t².

t terms: 8t - 3t - t = 4t.

Constants: -2.

Thus: 2t² + 4t - 2 = 0 => Divide by 2: t² + 2t - 1 = 0.

Solutions: t = [-2 ± sqrt(4 + 4)] / 2 = [-2 ± sqrt(8)] / 2 = [-2 ± 2√2]/2 = -1 ± √2.

Since t > 0, t = -1 + √2 ≈ -1 + 1.414 ≈ 0.414.

Ah! So the critical point is at t = √2 - 1 ≈ 0.414, not at t = sqrt(3) - 1. This is different from before! So previously, when I was minimizing g(t), I found t = sqrt(3) - 1, but when minimizing f(t), the critical point is at t = √2 - 1.

This explains the discrepancy. Previously, I confused the functions to minimize. So let me correct this.

So the correct critical point is at t = √2 - 1. Now, let me compute f(t) at this value.

Let me set t = √2 - 1. Then:

1 + t = 1 + √2 - 1 = √2.

t(2 + t): Let me compute t = √2 - 1, so 2 + t = 2 + √2 - 1 = 1 + √2. Therefore, t(2 + t) = (√2 - 1)(1 + √2) = (√2)^2 - 1^2 = 2 - 1 = 1.

Therefore, f(t) = (1 + t)^2 / sqrt(t(2 + t)) = (√2)^2 / sqrt(1) = 2 / 1 = 2.

This matches the result from the Lagrange multipliers approach! So, the minimum value is 2, achieved at t = √2 - 1, which corresponds to x = z = a, y = ta = (√2 - 1)a. Let me compute a from the constraint.

From the constraint: a^4 * t(2 + t) = 1. Since t(2 + t) = 1, this implies a^4 = 1, so a = 1 (since a > 0). Therefore, x = z = 1, y = t * a = (√2 - 1) * 1 = √2 - 1, which is the same as the result from the Lagrange multipliers method.

Therefore, the minimum value of (x + y)(y + z) is 2.

This resolves the discrepancy. Earlier, I incorrectly derived the critical point for g(t) instead of f(t), leading to confusion. Now, with the correct derivative calculation, we find that the minimum occurs at t = √2 - 1, leading to the minimum value of 2.

Therefore, the minimum value is 2, achieved when x = z = 1 and y = √2 - 1.

To confirm, let's check with the earlier substitution approach: when t = √2 - 1, a = 1, so x = 1, y = √2 - 1, z = 1, then (x + y)(y + z) = (1 + √2 - 1)(√2 - 1 + 1) = √2 * √2 = 2, which matches.

Thus, after correcting the earlier mistakes in the substitution approach, the minimum value is indeed 2.

To ensure there are no other minima when x ≠ z, we can consider that the Lagrange multipliers method found this minimum, and due to the symmetry of the problem, it's likely that this is the global minimum.

Therefore, the minimum value of (x + y)(y + z) is 2.

**Final Answer**
\boxed{2}

To find the **minimum value** of the expression
$$
(x + y)(y + z)
$$
given the constraint
$$
xyz(x + y + z) = 1
$$
with $x, y, z > 0$, we proceed by leveraging symmetry and optimal substitution.

---

### Step 1: Use Symmetry and Substitution

Assume that $x = z$, based on the symmetry in the constraint and the expression to minimize. This helps reduce the number of variables and simplifies the analysis.

Let $x = z = a$, and $y = b$. Then the constraint becomes:
$$
a^2 b (2a + b) = 1
$$
The objective function becomes:
$$
(x + y)(y + z) = (a + b)^2
$$
So the problem reduces to minimizing:
$$
(a + b)^2 \quad \text{given} \quad a^2 b (2a + b) = 1
$$

---

### Step 2: Introduce a New Variable

Let $t = \frac{b}{a}$, so $b = ta$. Then:
$$
a^2 (ta)(2a + ta) = t a^3 (2 + t) = 1 \Rightarrow a^3 = \frac{1}{t(2 + t)}
$$

So:
$$
a = \left(\frac{1}{t(2 + t)}\right)^{1/3}
$$

Then:
$$
(a + b)^2 = (a(1 + t))^2 = \left(\left(\frac{1}{t(2 + t)}\right)^{1/3}(1 + t)\right)^2 = \frac{(1 + t)^2}{[t(2 + t)]^{2/3}}
$$

So we need to minimize:
$$
f(t) = \frac{(1 + t)^2}{[t(2 + t)]^{2/3}}
\quad \text{for} \quad t > 0
$$

---

### Step 3: Minimize Using Calculus

Take the derivative of $f(t)$, or simplify by analyzing critical points. A more direct and accurate route is to use **Lagrange Multipliers**.

---

### Step 4: Apply Lagrange Multipliers

Let the objective function be:
$$
f(x, y, z) = (x + y)(y + z)
$$
and constraint:
$$
g(x, y, z) = xyz(x + y + z) - 1 = 0
$$

Compute partial derivatives:
- $f_x = y + z$
- $f_y = x + 2y + z$
- $f_z = x + y$

- $g_x = yz(x + y + z) + xyz$
- $g_y = xz(x + y + z) + xyz$
- $g_z = xy(x + y + z) + xyz$

Set up the system:
$$
f_x = \lambda g_x, \quad f_y = \lambda g_y, \quad f_z = \lambda g_z
$$

Assume $x = z$, and consider the symmetry. After solving the system, we find that the minimum occurs when:
$$
z = x = 1, \quad y = \sqrt{2} - 1
$$

Substitute these into the expression:
$$
(x + y)(y + z) = (1 + (\sqrt{2} - 1))(\sqrt{2} - 1 + 1) = \sqrt{2} \cdot \sqrt{2} = 2
$$

---

### Step 5: Verify Constraint is Satisfied

Check the original constraint:
$$
xyz(x + y + z) = 1 \cdot (\sqrt{2} - 1) \cdot 1 \cdot (1 + (\sqrt{2} - 1) + 1) = (\sqrt{2} - 1)(1 + \sqrt{2}) = 2 - 1 = 1
$$
✅ Constraint is satisfied.

---

### Final Answer

$$
\boxed{2}
$$

[Assistant]
Explanation:
Okay, so I need to find the minimum value of (x + y)(y + z) given that x, y, z are positive real numbers and satisfy the condition xyz(x + y + z) = 1. Hmm, let's see.

First, I should probably try to understand the problem and see if I can relate the given condition to the expression we need to minimize. The expression (x + y)(y + z) can be expanded, maybe that will help? Let me try expanding it:

(x + y)(y + z) = xy + xz + y² + yz.

Hmm, not sure if that helps directly. Maybe I can think of it in terms of variables. Let me note that the given condition is xyz(x + y + z) = 1. So, maybe I need to use some inequalities here? Like AM-GM inequality? Since we have products and sums, that might be useful.

But before that, perhaps I can make substitutions to reduce the number of variables? Let me think. The expression we need to minimize is (x + y)(y + z). Let me denote some variables. Let me think about symmetry. Is there a way to make some substitution that can simplify things?

Alternatively, maybe we can fix some variables? For example, maybe set some variables equal? Let me try to assume that x = z? Because sometimes symmetry gives the extremum. If x = z, maybe that would simplify things?

Let me try that. Suppose x = z. Then, let me denote x = z = a, and y = b. Then, the condition becomes:

xyz(x + y + z) = a * b * a * (a + b + a) = a²b(2a + b) = 1.

So, a²b(2a + b) = 1.

And the expression to minimize is (x + y)(y + z) = (a + b)(b + a) = (a + b)².

Oh! Wait, if x = z = a and y = b, then (x + y)(y + z) = (a + b)(b + a) = (a + b)^2. So, we need to minimize (a + b)^2, given that a²b(2a + b) = 1.

So, maybe this substitution can help? Let me think. Let me denote S = a + b, and maybe express the condition in terms of S and something else? Or maybe express in terms of variables?

Alternatively, maybe I can use Lagrange multipliers here? But since I'm trying to find a minimum, maybe there is a smarter way.

Alternatively, maybe set t = b/a? Let me try. Let me set t = b/a, so that b = ta. Then, substituting into the condition:

a² * ta * (2a + ta) = a² * ta * a(2 + t) = a^3 * t(2 + t) = 1.

Therefore, a^3 = 1/(t(2 + t)), so a = [1/(t(2 + t))]^{1/3}.

Then, S = a + b = a + ta = a(1 + t) = [1/(t(2 + t))]^{1/3} * (1 + t).

Therefore, (a + b)^2 = [ (1 + t)^3 / (t(2 + t)) ]^{2/3}.

So, to minimize (a + b)^2, we need to minimize [ (1 + t)^3 / (t(2 + t)) ]^{2/3} which is equivalent to minimizing (1 + t)^3 / (t(2 + t)).

Therefore, if I can find the minimum of f(t) = (1 + t)^3 / [t(2 + t)] for t > 0, then take that value and raise it to the 2/3 power, that will be the minimum of (a + b)^2.

So, let me focus on minimizing f(t) = (1 + t)^3 / [t(2 + t)] for t > 0.

To find the minimum of f(t), take derivative and set to zero. Let me compute f'(t). Let me denote numerator as N(t) = (1 + t)^3 and denominator as D(t) = t(2 + t) = 2t + t^2. Therefore, f(t) = N(t)/D(t).

So, f'(t) = [N'(t) D(t) - N(t) D'(t)] / [D(t)]².

First compute N'(t) = 3(1 + t)^2.

D'(t) = 2 + 2t. Wait, D(t) is t(2 + t) = 2t + t², so derivative is 2 + 2t? Wait, no. Wait, derivative of t(2 + t) is derivative of 2t + t² which is 2 + 2t. Yes, correct.

Therefore, f'(t) = [3(1 + t)^2 * (2t + t²) - (1 + t)^3 * (2 + 2t)] / (2t + t²)^2.

Let me factor out common terms. Let me note that numerator is:

(1 + t)^2 [3(2t + t²) - (1 + t)(2 + 2t)].

Let me compute the bracket:

First term: 3(2t + t²) = 6t + 3t²

Second term: (1 + t)(2 + 2t) = 2(1 + t)(1 + t) = 2(1 + t)^2 = 2(1 + 2t + t²) = 2 + 4t + 2t²

Therefore, the bracket is:

[6t + 3t²] - [2 + 4t + 2t²] = 6t + 3t² - 2 - 4t - 2t² = (6t - 4t) + (3t² - 2t²) - 2 = 2t + t² - 2.

So, numerator is (1 + t)^2 (t² + 2t - 2).

Therefore, critical points when numerator is zero. Since (1 + t)^2 is always positive for t > 0, we set t² + 2t - 2 = 0. Solve for t:

t = [-2 ± sqrt(4 + 8)] / 2 = [-2 ± sqrt(12)] / 2 = [-2 ± 2*sqrt(3)] / 2 = -1 ± sqrt(3).

Since t > 0, we take the positive root: t = -1 + sqrt(3). Wait, sqrt(3) is approx 1.732, so -1 + 1.732 ≈ 0.732 > 0. So, critical point at t = sqrt(3) - 1.

So, this is the critical point. Now we need to check if it's a minimum.

Let me check the second derivative or test intervals. Let me pick t approaching 0 from the right: for t approaching 0+, the numerator of f(t) is (1 + t)^3 ~ 1, and denominator t(2 + t) ~ 2t, so f(t) ~ 1/(2t) which tends to infinity. Similarly, as t approaches infinity, f(t) ~ t^3 / (t * t) = t^3 / t² = t, which tends to infinity. Therefore, the function tends to infinity at both ends, so the critical point must be a minimum.

Therefore, the minimum occurs at t = sqrt(3) - 1.

Now compute f(t) at t = sqrt(3) - 1:

First, compute t = sqrt(3) - 1.

Let me compute 1 + t = 1 + sqrt(3) - 1 = sqrt(3).

Compute denominator t(2 + t):

First compute 2 + t = 2 + sqrt(3) - 1 = 1 + sqrt(3).

Therefore, denominator is t*(1 + sqrt(3)) = (sqrt(3) - 1)*(1 + sqrt(3)) = (sqrt(3))^2 - 1^2 = 3 - 1 = 2.

Therefore, f(t) = (sqrt(3))^3 / 2.

Wait, numerator is (1 + t)^3 = (sqrt(3))^3 = 3*sqrt(3).

So, f(t) = 3√3 / 2.

Therefore, the minimal value of (a + b)^2 is [f(t)]^{2/3} = [ (3√3)/2 ]^{2/3}?

Wait, hold on. Wait, earlier we had (a + b)^2 = [ (1 + t)^3 / (t(2 + t)) ]^{2/3} = [f(t)]^{2/3}?

Wait, no. Wait, actually, earlier I had:

(a + b)^2 = [ (1 + t)^3 / (t(2 + t)) ]^{2/3} ?

Wait, let me retrace. Earlier:

We had S = a + b = [1/(t(2 + t))]^{1/3} * (1 + t). Therefore, (a + b)^2 = [ (1 + t)^3 / (t(2 + t)) ]^{2/3} ?

Wait, let me check:

If S = [1/(t(2 + t))]^{1/3} * (1 + t), then S^2 = [ (1 + t)^2 / (t(2 + t)) ]^{1/3} ?

Wait, no:

Wait, S = [ (1 + t) / (t(2 + t))^{1/3} ] ?

Wait, original substitution was that a^3 = 1 / [ t(2 + t) ], so a = [1 / (t(2 + t))]^{1/3}. Then S = a(1 + t) = [1 / (t(2 + t))]^{1/3} * (1 + t). Therefore, S^2 = [ (1 + t)^2 / (t(2 + t)) ]^{1/3} ?

Wait, so S^2 = [ (1 + t)^2 / (t(2 + t)) ]^{1/3} ?

But earlier I thought that (a + b)^2 is [ (1 + t)^3 / (t(2 + t)) ]^{2/3}... Wait, maybe I made a miscalculation earlier.

Let me redo that part. Let me clarify.

Original expression for S = a + b.

We had:

a^3 = 1 / [ t(2 + t) ]

Therefore, a = [1 / (t(2 + t))]^{1/3}

Then S = a(1 + t) = [1 / (t(2 + t))]^{1/3} * (1 + t)

Therefore, S^2 = [ (1 + t)^2 / (t(2 + t)) ]^{1/3}

Wait, so S^2 is equal to [ (1 + t)^2 / (t(2 + t)) ]^{1/3}

But earlier I thought that it was [ (1 + t)^3 / (t(2 + t)) ]^{2/3}, but that seems incorrect.

Wait, so if f(t) = (1 + t)^3 / [t(2 + t)], then [f(t)]^{2/3} would be [ (1 + t)^3 / (t(2 + t)) ]^{2/3} = (1 + t)^2 / [t(2 + t)]^{1/3}, which is the same as S^2. Therefore, yes, S^2 = [f(t)]^{2/3}. Therefore, since we found that the minimal value of f(t) is 3√3 / 2, then the minimal value of S^2 is [3√3 / 2]^{2/3}?

Wait, let me compute that:

Let me write 3√3 as 3 * 3^{1/2} = 3^{3/2}, so 3√3 / 2 = 3^{3/2}/2. Therefore, [3^{3/2}/2]^{2/3} = (3^{3/2})^{2/3} / 2^{2/3} = 3^{(3/2 * 2/3)} / 2^{2/3} = 3^1 / 2^{2/3} = 3 / 2^{2/3}.

But 2^{2/3} is the cube root of 4, so 3 divided by cube root of 4. But is that the minimal value? Wait, but maybe there is a simpler form? Alternatively, maybe I made a mistake in calculation earlier?

Wait, let me check again:

We found that f(t) = (1 + t)^3 / [t(2 + t)] reaches its minimum at t = sqrt(3) - 1, and the minimal value is 3√3 / 2? Wait, earlier when I computed f(t) at t = sqrt(3) - 1, I had:

(1 + t)^3 / [t(2 + t)] = (sqrt(3))^3 / 2 = (3*sqrt(3))/2. Yes, that's correct. Therefore, f(t) is minimized at 3√3 / 2. Therefore, S^2 = [f(t)]^{2/3} = [ (3√3)/2 ]^{2/3} ?

Alternatively, maybe we can write this as something else. Let me compute [ (3√3)/2 ]^{2/3}:

First, note that 3√3 = 3^{3/2}, so (3^{3/2}/2)^{2/3} = 3^{(3/2)*(2/3)} / 2^{2/3} = 3^1 / 2^{2/3} = 3 / 2^{2/3}.

Alternatively, rationalizing, 2^{2/3} is cube root of 4, so it's 3 divided by cube root of 4. But is there a better way to express this?

Alternatively, maybe I made a wrong assumption by setting x = z? Maybe the minimum occurs when x ≠ z? I need to check if this assumption is valid.

Alternatively, maybe I should try using Lagrange multipliers for the original problem without assuming x = z.

Let me try that approach as a check. Let me define variables x, y, z > 0, with constraint xyz(x + y + z) = 1, and want to minimize (x + y)(y + z).

Let me denote the function to minimize as f(x, y, z) = (x + y)(y + z). The constraint is g(x, y, z) = xyz(x + y + z) - 1 = 0.

Using Lagrange multipliers, we set ∇f = λ∇g.

Compute gradients:

First, compute partial derivatives of f:

df/dx = derivative of (x + y)(y + z) with respect to x is (1)(y + z) = y + z.

df/dy = derivative with respect to y is (x + y)'*(y + z) + (x + y)*(y + z)' = (1)(y + z) + (x + y)(1) = y + z + x + y = x + 2y + z.

df/dz = derivative with respect to z is (x + y)*(1) = x + y.

Now compute partial derivatives of g:

First, expand g(x, y, z) = xyz(x + y + z) - 1. Let me write it as xyz(x + y + z) - 1.

Compute dg/dx: derivative of xyz(x + y + z) with respect to x.

Use product rule: derivative of xyz * (x + y + z) + xyz * derivative of (x + y + z).

Wait, actually, better to expand first:

xyz(x + y + z) = x^2 y z + x y^2 z + x y z^2.

So, dg/dx = 2xyz + y^2 z + y z^2.

Similarly, dg/dy = x^2 z + 2x y z + x z^2.

dg/dz = x^2 y + x y^2 + 2x y z.

Wait, let me check:

Alternatively, using product rule:

g = xyz(x + y + z). Let me denote u = xyz, v = x + y + z. Then dg/dx = du/dx * v + u * dv/dx.

du/dx = yz, dv/dx = 1. Therefore, dg/dx = yz(x + y + z) + xyz(1) = yz(x + y + z + x) = yz(2x + y + z). Wait, that seems different from expanding. Wait, maybe mistake here?

Wait, no:

Wait, if u = xyz, then du/dx = yz. Then dv/dx = derivative of (x + y + z) w.r. to x is 1. Therefore, dg/dx = yz*(x + y + z) + xyz*1 = yz(x + y + z) + xyz = yz(x + y + z + x) ??? Wait, no:

Wait, it's yz(x + y + z) + xyz. Let me factor yz:

yz(x + y + z + x) ? Wait, no. Let me compute:

yz(x + y + z) + xyz = yz(x + y + z + x) ? No, that would be adding yz*x + yz*y + yz*z + xyz. Wait, no. Let me compute:

Original terms:

yz(x + y + z) = yzx + yzy + yzz = xyz + y²z + yz²

Plus xyz from the other term: total is 2xyz + y²z + yz². Which matches the expansion. Therefore, dg/dx = 2xyz + y²z + yz². Similarly for others.

So, the gradient of g is:

dg/dx = 2xyz + y²z + yz²,

dg/dy = 2xyz + x²z + xz²,

dg/dz = 2xyz + x²y + xy².

Therefore, setting ∇f = λ∇g:

So,

y + z = λ(2xyz + y²z + yz²) ...(1)

x + 2y + z = λ(2xyz + x²z + xz²) ...(2)

x + y = λ(2xyz + x²y + xy²) ...(3)

Additionally, we have the constraint xyz(x + y + z) = 1 ...(4)

This seems complex. Maybe we can assume some symmetry here. For example, if we assume that x = z, as before, then maybe equations simplify?

Let me try that. Let me set x = z. Then, variables are x, y, x. Let me denote x = z.

Then, let me compute equations (1), (2), (3):

Equation (1): y + x = λ(2x y x + y²x + y x²) = λ(2x² y + x y² + x² y) = λ(3x² y + x y²).

Equation (2): x + 2y + x = λ(2x y x + x² x + x x²) = λ(2x² y + x³ + x³) = λ(2x² y + 2x³).

Equation (3): x + y = λ(2x y x + x² y + x y²) = λ(2x² y + x² y + x y²) = λ(3x² y + x y²).

Therefore, equations (1) and (3) are the same when x = z. Therefore, equations (1) and (3) are same, so we have:

From equation (1): y + x = λ(3x² y + x y²) = λ x y (3x + y).

From equation (2): 2x + 2y = λ(2x² y + 2x³) = 2λx² (y + x).

Therefore, let me denote equation (1): x + y = λ x y (3x + y).

Equation (2): 2(x + y) = 2λ x² (x + y).

Assuming x + y ≠ 0 (which it isn't, since x, y > 0), we can divide both sides by (x + y):

From equation (2): 2 = 2λ x² => λ = 1 / x².

From equation (1): x + y = (1 / x²) * x y (3x + y) => x + y = (y / x)(3x + y).

Multiply both sides by x:

x(x + y) = y(3x + y)

Expand left side: x² + x y

Right side: 3x y + y²

Bring all terms to left:

x² + x y - 3x y - y² = x² - 2x y - y² = 0.

So:

x² - 2x y - y² = 0

Let me write it as x² - 2x y - y² = 0.

This is quadratic in x: x² - 2y x - y² = 0.

Solving for x:

x = [2y ± sqrt(4y² + 4y²)] / 2 = [2y ± sqrt(8y²)] / 2 = [2y ± 2y√2]/2 = y(1 ± √2).

Since x > 0 and y > 0, we discard the negative solution, so x = y(1 + √2). Therefore, x = y(1 + √2).

So, with x = z = y(1 + √2). Let me denote this as x = z = k y, where k = 1 + √2.

Now, let's use the constraint equation (4): xyz(x + y + z) = 1.

Substituting x = z = k y:

(k y) * y * (k y) * (k y + y + k y) = k² y³ * ( (2k + 1) y ) = k² (2k + 1) y^4 = 1.

Therefore, y^4 = 1 / [k² (2k + 1)].

Compute k = 1 + √2 ≈ 2.414, but let me keep it exact.

First compute 2k + 1 = 2(1 + √2) + 1 = 2 + 2√2 + 1 = 3 + 2√2.

Therefore, y^4 = 1 / [k² (3 + 2√2)].

But k = 1 + √2. Let me compute k²:

k² = (1 + √2)^2 = 1 + 2√2 + 2 = 3 + 2√2.

Therefore, k² (3 + 2√2) = (3 + 2√2)^2.

Compute (3 + 2√2)^2 = 9 + 12√2 + 8 = 17 + 12√2.

Therefore, y^4 = 1 / (17 + 12√2).

But perhaps we can rationalize the denominator? Let me see:

Multiply numerator and denominator by (17 - 12√2):

Wait, but denominator is 17 + 12√2. Let me check:

Wait, but wait, y^4 = 1 / [ (3 + 2√2)^2 ] = 1 / [ (sqrt(3 + 2√2))^4 ]... Wait, but 3 + 2√2 is (1 + √2)^2, so (3 + 2√2)^2 = (1 + √2)^4. Therefore, y^4 = 1 / (1 + √2)^4, so y = [1 / (1 + √2)^4]^{1/4} = 1 / (1 + √2).

Therefore, y = 1 / (1 + √2). Rationalizing:

Multiply numerator and denominator by (√2 - 1):

y = (√2 - 1)/[(1 + √2)(√2 - 1)] = (√2 - 1)/(2 - 1) = √2 - 1.

So y = √2 - 1. Then, x = z = k y = (1 + √2)(√2 - 1) = (√2)^2 - 1^2 = 2 - 1 = 1.

So x = z = 1, y = √2 - 1.

Therefore, let me check the value of (x + y)(y + z). Since x = z = 1, and y = √2 - 1.

Compute (x + y) = 1 + √2 - 1 = √2.

Similarly, (y + z) = √2 - 1 + 1 = √2. Therefore, (x + y)(y + z) = √2 * √2 = 2.

Wait, so the minimal value is 2? But earlier when I did the substitution with x = z, I found that the minimal value was 3 / 2^{2/3} ≈ 3 / 1.587 ≈ 1.89, which is less than 2. But that contradicts this result? Which one is correct?

Wait, there must be a mistake in my previous reasoning. Let me check.

Earlier, when I set x = z, then found that (a + b)^2 is [f(t)]^{2/3} where f(t) had a minimum of 3√3 / 2. Then, [ (3√3)/2 ]^{2/3}... Let me compute that numerically.

Let me compute 3√3: √3 ≈ 1.732, so 3*1.732 ≈ 5.196. Divided by 2: ≈ 2.598. Then take 2/3 power: 2.598^(2/3). Let me compute ln(2.598) ≈ 0.955. Multiply by 2/3: ≈ 0.637. Exponentiate: e^0.637 ≈ 1.89. But according to the Lagrange multipliers approach, the minimal value is 2. Which is higher. Therefore, there must be an error in one of the approaches.

But since Lagrange multipliers gave us an exact value of 2, and the substitution approach gave us approx 1.89, which is lower, but maybe I made a mistake in the substitution approach?

Wait, but when using the Lagrange multipliers with x = z, we found that the minimal value is 2. But if there exists another case where x ≠ z which gives a lower value, then that approach would be wrong. However, since Lagrange multipliers should account for all possibilities, perhaps there was a miscalculation in substitution approach?

Let me check substitution approach again.

Earlier, when I set x = z = a, y = b, then S = a + b. The expression to minimize was S^2 = [ (1 + t)^2 / (t(2 + t)) ]^{1/3} where t = b/a. Then I tried to minimize f(t) = (1 + t)^3 / [ t(2 + t) ] and found its minimum at t = sqrt(3) - 1, leading to S^2 = [3√3 / 2]^{2/3} ≈ 1.89. However, according to the Lagrange multipliers, when x = z, the minimum is achieved at 2. Therefore, these two results are conflicting. Which is correct?

Wait, let me check with actual numbers. If in the Lagrange approach, with x = z = 1, y = √2 - 1 ≈ 0.414, then let's compute the original condition xyz(x + y + z) = 1 * (√2 - 1) * 1 * (1 + (√2 - 1) + 1) = (√2 - 1) * (1 + √2 - 1 + 1) = (√2 - 1)(1 + √2) = (√2 - 1)(√2 + 1) = 2 - 1 = 1. Correct. So that satisfies the condition. And (x + y)(y + z) = 2.

Now, let me check with the substitution approach. If I take t = sqrt(3) - 1 ≈ 0.732. Then, let me compute b = t*a, and from earlier, a^3 = 1 / [ t(2 + t) ].

Let me compute t(2 + t): t ≈ 0.732, 2 + t ≈ 2.732. Multiply: 0.732 * 2.732 ≈ 2. So, a^3 ≈ 1/2, so a ≈ 0.7937. Then, b = t*a ≈ 0.732 * 0.7937 ≈ 0.581. Then, (x + y)(y + z) = (a + b)^2 ≈ (0.7937 + 0.581)^2 ≈ (1.3747)^2 ≈ 1.89, which is indeed less than 2. So according to this, there's a lower value. But why does the Lagrange multipliers method give a higher value?

Ah! Because in the Lagrange multipliers approach, when I assumed x = z, I found a critical point, but perhaps there is another critical point when x ≠ z, which gives a lower value. However, in the Lagrange multipliers approach, we set up equations and found that assuming x = z leads to a critical point, but maybe there are other critical points with x ≠ z. However, since the substitution approach with x = z led to a lower value, but the Lagrange multipliers didn't consider that? Or maybe I made a mistake in the substitution approach?

Wait, no. The substitution approach with x = z is a special case, so the minimal value could be either in that case or in another case. Therefore, the Lagrange multipliers method, when assuming x = z, gives a critical point with value 2, but there's another critical point (with x ≠ z) that gives a lower value. However, perhaps my mistake is in the substitution approach. Let me check again.

Wait, in the substitution approach, we set x = z and found that the minimal value of (x + y)(y + z) was approximately 1.89, but when I calculated with Lagrange multipliers with x = z, I get 2. So there is inconsistency. Therefore, I must have made a mistake in one of the steps.

Let me check the substitution approach again.

Earlier, I had:

After setting x = z = a, y = b, then the constraint is a²b(2a + b) = 1. Then, we set t = b/a, so b = t a. Therefore, the constraint becomes a² * t a * (2a + t a) = a³ t (2 + t) = 1 => a³ = 1/(t(2 + t)) => a = [1/(t(2 + t))]^{1/3}.

Then, (x + y)(y + z) = (a + b)^2 = [a(1 + t)]². So [a(1 + t)]² = [ (1 + t)^2 / (t(2 + t)) ]^{2/3} ?

Wait, hold on. If a = [1/(t(2 + t))]^{1/3}, then [a(1 + t)]² is [ (1 + t)^2 / (t(2 + t))^{1/3} ] ?

Wait, no. Wait:

a(1 + t) = [1/(t(2 + t))]^{1/3} * (1 + t)

Therefore, [a(1 + t)]² = [ (1 + t)^2 / (t(2 + t))^{1/3} ] ?

Wait, no. Let me compute:

[a(1 + t)]^2 = [ (1 + t) / (t(2 + t))^{1/3} ]² = (1 + t)^2 / [ t(2 + t) ]^{2/3}

But earlier I thought it was [ (1 + t)^3 / (t(2 + t)) ]^{2/3}, which is different. So I had an error in my substitution approach.

So the correct expression is (a + b)^2 = (1 + t)^2 / [ t(2 + t) ]^{2/3}

Therefore, to minimize this expression, we need to minimize f(t) = (1 + t)^2 / [ t(2 + t) ]^{2/3}

But previously, I incorrectly thought I needed to minimize (1 + t)^3 / [ t(2 + t) ], which led to a different result. So that was my mistake! Therefore, let me correct this.

So, to minimize f(t) = (1 + t)^2 / [ t(2 + t) ]^{2/3} over t > 0.

Let me denote this as f(t) = [(1 + t)^2] / [t(2 + t)]^{2/3}. To minimize this, take natural logarithm to make differentiation easier.

Let me set ln f(t) = 2 ln(1 + t) - (2/3)(ln t + ln(2 + t)).

Take derivative with respect to t:

d/dt [ln f(t)] = 2/(1 + t) - (2/3)[1/t + 1/(2 + t)].

Set derivative to zero:

2/(1 + t) - (2/3)[1/t + 1/(2 + t)] = 0.

Multiply both sides by 3 to eliminate denominators:

6/(1 + t) - 2[1/t + 1/(2 + t)] = 0.

Simplify:

6/(1 + t) = 2[1/t + 1/(2 + t)].

Divide both sides by 2:

3/(1 + t) = 1/t + 1/(2 + t).

Let me compute RHS: 1/t + 1/(2 + t) = [ (2 + t) + t ] / [ t(2 + t) ] = (2 + 2t)/[t(2 + t)] = 2(1 + t)/[t(2 + t)].

Therefore:

3/(1 + t) = 2(1 + t)/[t(2 + t)]

Multiply both sides by (1 + t):

3 = 2(1 + t)^2 / [t(2 + t)]

Multiply both sides by t(2 + t):

3t(2 + t) = 2(1 + t)^2.

Expand both sides:

Left side: 6t + 3t².

Right side: 2(1 + 2t + t²) = 2 + 4t + 2t².

Bring all terms to left:

6t + 3t² - 2 - 4t - 2t² = 0 => (6t - 4t) + (3t² - 2t²) - 2 = 2t + t² - 2 = 0.

So, t² + 2t - 2 = 0, which is the same equation as before! Therefore, t = [ -2 ± sqrt(4 + 8) ] / 2 = [ -2 ± sqrt(12) ] / 2 = -1 ± sqrt(3). As before, t = sqrt(3) - 1 ≈ 0.732.

Therefore, even after correcting my substitution approach, the critical point is same t = sqrt(3) - 1. Therefore, compute f(t) at this t:

f(t) = (1 + t)^2 / [ t(2 + t) ]^{2/3}.

First compute 1 + t = sqrt(3) as before. Then, t(2 + t) = (sqrt(3) - 1)(1 + sqrt(3)) = 3 - 1 = 2 as before. So [ t(2 + t) ] = 2. Therefore, f(t) = (sqrt(3))² / [2]^{2/3} = 3 / 2^{2/3} ≈ 3 / 1.587 ≈ 1.89.

But in the Lagrange multipliers approach, with x = z, I found that the minimal value is 2. So which one is correct?

Wait, let me check with actual numbers. Let me take t = sqrt(3) - 1 ≈ 0.732. Then, a³ = 1 / [t(2 + t)] = 1/2, so a ≈ (1/2)^{1/3} ≈ 0.7937. Then, b = t * a ≈ 0.732 * 0.7937 ≈ 0.581. Then, check if the original condition xyz(x + y + z) = 1:

x = a ≈ 0.7937, y ≈ 0.581, z = a ≈ 0.7937. Then, x + y + z ≈ 0.7937 + 0.581 + 0.7937 ≈ 2.168. Then, xyz(x + y + z) ≈ 0.7937 * 0.581 * 0.7937 * 2.168. Let me compute step by step:

First compute 0.7937 * 0.7937 ≈ 0.63, then multiply by 0.581: ≈ 0.63 * 0.581 ≈ 0.366. Then multiply by 2.168: ≈ 0.366 * 2.168 ≈ 0.793. Which is not 1. Wait, that's not equal to 1. That suggests that my substitution approach has an error?

Wait, no. Earlier, when I set x = z = a, y = b, then the constraint is a²b(2a + b) = 1. With t = b/a = sqrt(3) - 1, and a³ = 1/[t(2 + t)] = 1/2. Therefore, a = 2^{-1/3}. Then b = t * a = (sqrt(3) - 1) * 2^{-1/3}. Then, let me compute a²b(2a + b):

Compute a²b(2a + b) = a² * b * 2a + a² * b * b = 2a³b + a²b².

But given that a³ = 1/[t(2 + t)] = 1/2, and b = t a:

So 2a³b = 2*(1/2)*b = b. And a²b² = (a b)^2. But a b = a * t a = t a². Since a³ = 1/2, then a² = (1/2)^{2/3}, so a b = t * (1/2)^{2/3}.

Therefore, total expression: b + (t * (1/2)^{2/3})². But I need to check if this is equal to 1?

Wait, but according to the constraint, a²b(2a + b) = 1. Let me compute:

a²b(2a + b) = a²b*2a + a²b*b = 2a³b + a²b².

But since a³ = 1/(t(2 + t)) = 1/2, so 2a³b = 2*(1/2)*b = b.

Meanwhile, a²b² = (a b)^2 = [ a * b ]^2. But a b = a * t a = t a². Since a³ = 1/2, so a² = a^{-1} * a³ = a^{-1} * 1/2. Wait, perhaps better to compute numerically:

Given a³ = 1/2, so a = 2^{-1/3}, then a² = 2^{-2/3}, and b = t * 2^{-1/3}. Then, a²b² = [2^{-2/3} * t^2 * 2^{-2/3}] = t² * 2^{-4/3}.

Therefore, total expression: b + a²b² = t * 2^{-1/3} + t² * 2^{-4/3}. Let me plug t = sqrt(3) - 1 ≈ 0.732:

First term: 0.732 * 2^{-1/3} ≈ 0.732 * 0.7937 ≈ 0.581.

Second term: 0.732² * 2^{-4/3} ≈ 0.536 * 0.396 ≈ 0.212.

Total ≈ 0.581 + 0.212 ≈ 0.793, which is not 1. Wait, that contradicts the condition. Therefore, where is the mistake?

Wait, earlier I thought that with a³ = 1/[t(2 + t)] which is 1/2, so a²b(2a + b) = 1? But according to this, it is not. Therefore, my mistake was in the substitution approach. Where?

Originally, I said that with x = z = a, y = b, then the constraint is a²b(2a + b) = 1. Is this correct?

Yes, because xyz(x + y + z) = a * b * a * (a + b + a) = a²b(2a + b). So that is correct. Then, when I set t = b/a, then b = ta, and the constraint becomes a²*(ta)*(2a + ta) = a² * ta * a(2 + t) = a³ t(2 + t) = 1 => a³ = 1/[t(2 + t)]. Therefore, this is correct. Therefore, if we take t = sqrt(3) - 1, then a³ = 1/[t(2 + t)] = 1/2. Then, why does a²b(2a + b) not equal to 1? Let me compute it properly:

Compute a²b(2a + b):

= a²b*(2a + b)

But since a³ = 1/2 => 2a³ = 1 => 2a³ = 1. Also, since b = t a, then:

Let me compute a²b*(2a + b) = a² * t a * (2a + t a) = a³ * t (2 + t) = (1/2) * t(2 + t) = (1/2)*2 = 1.

Wait! Because t(2 + t) = 2? Earlier, when t = sqrt(3) - 1, we had t(2 + t) = 2. Yes! Therefore, a³ * t(2 + t) = (1/2)*2 = 1. So it does satisfy the constraint. Therefore, my previous calculation where I computed numerically was wrong. Let me check again with numbers.

Let me compute a³ = 1/2, so a ≈ 0.7937. Then, t = sqrt(3) - 1 ≈ 0.732. b = a*t ≈ 0.732 * 0.7937 ≈ 0.581. Then, x + y + z = a + b + a = 2a + b ≈ 2*0.7937 + 0.581 ≈ 1.5874 + 0.581 ≈ 2.1684. Then, xyz(x + y + z) = a * b * a * (x + y + z) = a²b*(x + y + z) ≈ (0.7937² * 0.581) * 2.1684 ≈ (0.63 * 0.581) * 2.1684 ≈ 0.366 * 2.1684 ≈ 0.793. Wait, but according to algebra, it should be 1. Why discrepancy?

Ah! Because I miscalculated the value of a. If a³ = 1/2, then a = (1/2)^{1/3} ≈ 0.7937, but let me use exact values. Let me calculate more accurately.

First, compute a³ = 1/2, so a = 2^{-1/3} ≈ 0.7937005259840998.

Then, t = sqrt(3) - 1 ≈ 1.7320508075688772 - 1 = 0.7320508075688772.

Then, b = a * t ≈ 0.7937005259840998 * 0.7320508075688772 ≈ let me compute:

First multiply 0.7937 * 0.73205:

Approximate:

0.7937 * 0.7 = 0.5556

0.7937 * 0.03205 ≈ 0.7937 * 0.03 ≈ 0.0238 + 0.7937 * 0.00205 ≈ 0.0238 + 0.001627 ≈ 0.0254

Total approx. 0.5556 + 0.0254 ≈ 0.581. So b ≈ 0.581.

Then, x + y + z = 2a + b ≈ 2*0.7937 + 0.581 ≈ 1.5874 + 0.581 ≈ 2.1684.

Now, compute xyz(x + y + z) = a² * b * (x + y + z).

Let me compute a² * b: a² = (2^{-1/3})² = 2^{-2/3} ≈ 0.63. Then, a² * b ≈ 0.63 * 0.581 ≈ 0.366. Then multiply by x + y + z ≈ 2.1684: 0.366 * 2.1684 ≈ 0.793. But according to algebra, it should be 1. Where is this discrepancy?

Wait, let me calculate exactly:

Since a³ = 1/2 and t(2 + t) = 2, then a³ * t(2 + t) = 1/2 * 2 = 1. Therefore, xyz(x + y + z) = 1. But why when I compute numerically it's about 0.793? Because of rounding errors in my approximate calculations. Let me try with more precise values.

Let me use exact expressions:

Let me compute a = 2^{-1/3}, b = t * a = (sqrt(3) - 1) * 2^{-1/3}.

Then, x + y + z = 2a + b = 2 * 2^{-1/3} + (sqrt(3) - 1) * 2^{-1/3} = [2 + sqrt(3) - 1] * 2^{-1/3} = [1 + sqrt(3)] * 2^{-1/3}.

Now compute xyz(x + y + z) = a² * b * (x + y + z) = a² * b * (x + y + z).

Compute step by step:

First, a² = 2^{-2/3}.

b = t * a = (sqrt(3) - 1) * 2^{-1/3}.

(x + y + z) = [1 + sqrt(3)] * 2^{-1/3}.

Multiply all: 2^{-2/3} * (sqrt(3) - 1) * 2^{-1/3} * [1 + sqrt(3)] * 2^{-1/3} =

Combine exponents of 2: -2/3 -1/3 -1/3 = -4/3.

Multiply the rest: (sqrt(3) - 1)(sqrt(3) + 1) = (sqrt(3))^2 - 1^2 = 3 - 1 = 2.

Therefore, total expression: 2 * 2^{-4/3} = 2^{1 - 4/3} = 2^{-1/3} ≈ 0.7937, which is not 1. Wait, this contradicts the earlier algebraic conclusion that it should be 1.

Wait, this is a problem. Earlier I thought a³ * t(2 + t) = 1, but it seems like it's not. Let me re-express:

Original constraint: a³ * t(2 + t) = 1.

But t(2 + t) = 2, so a³ * 2 = 1 => a³ = 1/2, which is correct. Therefore, xyz(x + y + z) = a²b(2a + b) = a²b(2a + b) = a³t(2 + t) = (1/2)*2 = 1.

But according to the calculation above, it's 2 * 2^{-4/3} = 2^{-1/3} ≈ 0.7937? What is wrong here?

Wait, no! Let me check again:

Wait, xyz(x + y + z) = a²b(2a + b). And we have:

a²b(2a + b) = a²b*2a + a²b*b = 2a³b + a²b².

But earlier, I thought this equals a³ * t(2 + t). But why?

Because I had: a²b(2a + b) = a² * b * (2a + b) = a³ * b * (2 + b/a) = a³ * b * (2 + t). But since b = t a, then yes, this is a³ * t a * (2 + t) = a^4 t (2 + t). Wait, this contradicts the earlier assertion. Wait, no, let me re-express:

Wait, I think I have a mistake here. Let me redo:

Original expression: a²b(2a + b) = a²b * 2a + a²b * b = 2a³b + a²b².

But since b = t a, substitute:

= 2a³ * t a + a² * (t a)^2 = 2 t a^4 + t² a^4 = a^4(2t + t²).

But earlier I thought this was a³ * t(2 + t). Where is the error?

Ah, no! Previously, I thought that a²b(2a + b) = a³ * t(2 + t). But that is incorrect. It should be a^4(2t + t²). Therefore, my previous assertion that a³ * t(2 + t) = 1 was wrong. It should be a^4(2t + t²) = 1. Therefore, my mistake was here. Earlier, I thought the constraint was a³ * t(2 + t) = 1, but actually it's a^4(2t + t²) = 1. Wait, where did that come from?

Let me go back. Original constraint: xyz(x + y + z) = 1. With x = a, y = b, z = a. Then:

xyz(x + y + z) = a * b * a * (a + b + a) = a²b(2a + b) = 1.

With b = t a, so this becomes a² * t a * (2a + t a) = t a³ * a(2 + t) = t a^4 (2 + t) = 1.

Wait, so the constraint is t a^4 (2 + t) = 1, not a³ * t(2 + t) = 1. Earlier I had written a³ = 1/[t(2 + t)], which was wrong. That was the mistake!

Therefore, correcting that:

From the constraint: t a^4 (2 + t) = 1 => a^4 = 1 / [t(2 + t)] => a = [1 / (t(2 + t))]^{1/4}

Therefore, previous substitution approach had a mistake in the exponent of a.

Therefore, S = a + b = a + t a = a(1 + t) = [1 / (t(2 + t))]^{1/4} * (1 + t)

Thus, (x + y)(y + z) = (a + b)^2 = [ (1 + t)^2 / (t(2 + t)) ]^{1/2}

Wait, raising to power 1/2?

Yes, because S = [1 / (t(2 + t))]^{1/4} * (1 + t), so squaring it gives [ (1 + t)^2 / (t(2 + t)) ]^{1/2}

So, (x + y)(y + z) = sqrt( (1 + t)^2 / [ t(2 + t) ]) = (1 + t) / sqrt(t(2 + t))

Therefore, now, the expression to minimize is f(t) = (1 + t) / sqrt(t(2 + t)) for t > 0.

Previously, I had different exponents due to miscalculations. Now, this is the correct expression.

So, now, let's minimize f(t) = (1 + t)/sqrt(t(2 + t)). Let me square it to make it easier, since sqrt is increasing. Let me define g(t) = [f(t)]² = (1 + t)^2 / [t(2 + t)].

Wait, this is the same function f(t) that I incorrectly minimized earlier, but now with correct understanding, it's the same function as before. Wait, earlier when I corrected the substitution approach, I had this expression and found the critical point at t = sqrt(3) - 1.

So, let me re-express: the expression to minimize is g(t) = (1 + t)^2 / [t(2 + t)]. Earlier, I had found that its minimum is 3√3 / 2 at t = sqrt(3) - 1. Therefore, [f(t)]² = g(t), so the minimum of f(t) is sqrt(3√3 / 2). Wait, no. Wait, if g(t) has minimum value 3√3 / 2, then f(t) = sqrt(g(t)) has minimum value sqrt(3√3 / 2).

Wait, let me compute this value:

First, 3√3 / 2 is approximately 3 * 1.732 / 2 ≈ 5.196 / 2 ≈ 2.598. Then sqrt(2.598) ≈ 1.612.

But earlier, in the Lagrange multipliers approach, with x = z, I found that the minimal value was 2. Now, with corrected substitution approach, I find that it's approx 1.612, which is less than 2. So, this suggests that there is a lower value.

But let's check with exact values. At t = sqrt(3) - 1, we have:

g(t) = (1 + t)^2 / [t(2 + t)] = (sqrt(3))^2 / 2 = 3 / 2.

Wait, earlier I thought it was 3√3 / 2, but no. Wait, let me compute:

If t = sqrt(3) - 1, then 1 + t = sqrt(3), and t(2 + t) = 2. Therefore, g(t) = (sqrt(3))² / 2 = 3 / 2. Therefore, f(t) = sqrt(g(t)) = sqrt(3/2) ≈ 1.2247.

Wait, this is different from my previous calculation. What is going on?

Wait, previously, I thought that g(t) = (1 + t)^2 / [t(2 + t)] had a minimum of 3√3 / 2, but no, with t = sqrt(3) - 1, g(t) = 3 / 2. Wait, how did I get 3√3 / 2 before? That was a mistake. Let me retrace.

Earlier, when I was thinking of f(t) = (1 + t)^3 / [t(2 + t)], then at t = sqrt(3) - 1, it was (sqrt(3))^3 / 2 = 3√3 / 2. But that was for a different function. Now, with the corrected expression, g(t) = (1 + t)^2 / [t(2 + t)] has minimum value 3/2 at t = sqrt(3) - 1.

Therefore, the expression (x + y)(y + z) is sqrt(g(t)) = sqrt(3/2) ≈ 1.2247? Wait, no. Wait, the expression to minimize was f(t) = (1 + t)/sqrt(t(2 + t)) = sqrt(g(t)), and g(t) has minimum value 3/2, so f(t) has minimum value sqrt(3/2). Wait, sqrt(3/2) ≈ 1.2247.

But earlier, with the Lagrange multipliers approach, assuming x = z, we found that the minimal value is 2. How to resolve this?

Let me check with t = sqrt(3) - 1 ≈ 0.732, then compute (x + y)(y + z) = f(t) = (1 + t)/sqrt(t(2 + t)) = sqrt(3)/sqrt(2) = sqrt(3/2) ≈ 1.2247.

Let me verify this with the actual values.

With t = sqrt(3) - 1, and a^4 = 1 / [t(2 + t)] = 1/2, so a = (1/2)^{1/4} = 2^{-1/4} ≈ 0.8409. Then, b = t * a ≈ 0.732 * 0.8409 ≈ 0.616. Then, x + y + z = 2a + b ≈ 2*0.8409 + 0.616 ≈ 1.6818 + 0.616 ≈ 2.2978.

Now, compute xyz(x + y + z) = a² * b * (x + y + z) ≈ (0.8409²) * 0.616 * 2.2978. Compute step by step:

0.8409² ≈ 0.707.

Multiply by 0.616: 0.707 * 0.616 ≈ 0.435.

Multiply by 2.2978: 0.435 * 2.2978 ≈ 1.000. Yes! It satisfies the constraint.

Now, compute (x + y)(y + z). Since x = a ≈ 0.8409, y ≈ 0.616, z = a ≈ 0.8409. Then, (x + y)(y + z) = (a + y)(y + a) = (a + y)^2 ≈ (0.8409 + 0.616)^2 ≈ (1.4569)^2 ≈ 2.123. Wait, but sqrt(3/2) ≈ 1.2247, which is different. What is happening here?

Wait, no! Wait, my expression for (x + y)(y + z) is (a + b)^2, where b = y. But in reality, with x = a, y = b, z = a, (x + y)(y + z) = (a + b)(b + a) = (a + b)^2. But according to the previous substitution approach, this equals f(t) = sqrt(g(t)) = sqrt(3/2). But calculations show approximately 2.123. Which is different from sqrt(3/2) ≈ 1.2247.

This suggests another error in substitution approach. Where is the mistake?

Let me re-express: we had (x + y)(y + z) = (a + b)^2 = [a(1 + t)]^2 = a²(1 + t)^2.

But with a^4 = 1 / [t(2 + t)] => a² = 1 / [t(2 + t)]^{1/2}.

Therefore, (x + y)(y + z) = [1 / sqrt(t(2 + t))] * (1 + t)^2 = (1 + t)^2 / sqrt(t(2 + t)) = f(t).

But earlier, I wrote that this is equal to sqrt(g(t)), where g(t) = (1 + t)^2 / [t(2 + t)]. But actually, f(t) = g(t)^{1/2} * something? Wait, no:

Wait, g(t) = (1 + t)^2 / [t(2 + t)].

So f(t) = (1 + t)^2 / sqrt(t(2 + t)) = g(t) * sqrt(t(2 + t)) / sqrt(t(2 + t)) ? Wait, no:

Wait, f(t) = (1 + t)^2 / [t(2 + t)]^{1/2} = g(t)^{1/2} * [t(2 + t)]^{1/2} ? No, let me see:

Actually, g(t) is (1 + t)^2 / [t(2 + t)], and f(t) is (1 + t)^2 / sqrt(t(2 + t)) = g(t) * sqrt(t(2 + t)).

Wait, no, it's f(t) = g(t) * sqrt(t(2 + t))? Let me see:

Wait, g(t) = (1 + t)^2 / [t(2 + t)].

If I multiply g(t) by sqrt(t(2 + t)), I get [(1 + t)^2 / [t(2 + t)] ] * sqrt(t(2 + t)) = (1 + t)^2 / sqrt(t(2 + t)) = f(t).

So f(t) = g(t) * sqrt(t(2 + t)).

But earlier, when I computed g(t) at t = sqrt(3) - 1, it was 3/2. And sqrt(t(2 + t)) = sqrt(2). Therefore, f(t) = (3/2) * sqrt(2) ≈ 1.5 * 1.414 ≈ 2.121, which matches the numerical calculation of ≈ 2.123.

Ah! So my previous mistake was in thinking that f(t) was sqrt(g(t)), but it is actually g(t) * sqrt(t(2 + t)). Therefore, the expression to minimize is f(t) = g(t) * sqrt(t(2 + t)) = [(1 + t)^2 / [t(2 + t)]] * sqrt(t(2 + t)) = (1 + t)^2 / [t(2 + t)]^{1/2}.

Therefore, to find its minimum, we need to minimize this expression. However, earlier, I had found the critical point of g(t), which is (1 + t)^2 / [t(2 + t)], and found that its minimum is 3/2 at t = sqrt(3) - 1. But since f(t) = g(t) * sqrt(t(2 + t)), we need to minimize this product.

Wait, but how does this relate to the previous derivative calculation? Earlier, when I set up the derivative for g(t), I found that its minimum occurs at t = sqrt(3) - 1. Let me see what f(t) is at that point:

f(t) = g(t) * sqrt(t(2 + t)) = (3/2) * sqrt(2) ≈ 2.121, which matches the numerical calculation. So the minimum of f(t) is (3/2)*sqrt(2) = 3√2 / 2 ≈ 2.121. However, in the Lagrange multipliers approach with x = z, I found the value of (x + y)(y + z) as 2, which is less than 2.121. This is contradictory.

Wait, but earlier when I used the Lagrange multipliers with x = z, I found that the minimal value is 2, but according to this substitution approach, it's approximately 2.121. So which one is correct?

Let me check with the exact values from the Lagrange multipliers approach. When x = z = 1, y = sqrt(2) - 1 ≈ 0.414, then (x + y)(y + z) = (1 + y)^2. Let me compute it:

1 + y = 1 + sqrt(2) - 1 = sqrt(2), so (sqrt(2))^2 = 2.

But according to the substitution approach, when t = sqrt(3) - 1 ≈ 0.732, the value is approximately 2.121. Which is higher than 2. So this suggests that the minimum is 2, achieved at x = z = 1, y = sqrt(2) - 1.

But why does the substitution approach suggest a lower value? Because I think I made an error in interpreting the critical point. Earlier, I was trying to minimize g(t) = (1 + t)^2 / [t(2 + t)], but actually, the expression to minimize is f(t) = (1 + t)^2 / sqrt(t(2 + t)) = g(t) * sqrt(t(2 + t)).

So, to find the minimum of f(t), we cannot just minimize g(t), but need to consider the whole expression. Therefore, let me go back to the derivative.

Let me define f(t) = (1 + t)^2 / sqrt(t(2 + t)) = (1 + t)^2 / [t(2 + t)]^{1/2}. Let me take the derivative of f(t) with respect to t and set it to zero.

Let me set f(t) = (1 + t)^2 * [t(2 + t)]^{-1/2}.

Take logarithm: ln f(t) = 2 ln(1 + t) - (1/2) ln(t) - (1/2) ln(2 + t).

Take derivative:

f'(t)/f(t) = 2/(1 + t) - (1/2)(1/t) - (1/2)(1/(2 + t)).

Set to zero:

2/(1 + t) - 1/(2t) - 1/(2(2 + t)) = 0.

Multiply all terms by 2t(2 + t)(1 + t) to eliminate denominators:

2 * 2t(2 + t) - (1 + t)(2 + t) - (1 + t)t = 0.

Simplify term by term:

First term: 4t(2 + t) = 8t + 4t².

Second term: -(1 + t)(2 + t) = -(2 + t + 2t + t²) = - (2 + 3t + t²).

Third term: -(1 + t)t = -t - t².

Combine all terms:

8t + 4t² - 2 - 3t - t² - t - t² = 0.

Simplify:

Combine like terms:

t² terms: 4t² - t² - t² = 2t².

t terms: 8t - 3t - t = 4t.

Constants: -2.

Thus: 2t² + 4t - 2 = 0 => Divide by 2: t² + 2t - 1 = 0.

Solutions: t = [-2 ± sqrt(4 + 4)] / 2 = [-2 ± sqrt(8)] / 2 = [-2 ± 2√2]/2 = -1 ± √2.

Since t > 0, t = -1 + √2 ≈ -1 + 1.414 ≈ 0.414.

Ah! So the critical point is at t = √2 - 1 ≈ 0.414, not at t = sqrt(3) - 1. This is different from before! So previously, when I was minimizing g(t), I found t = sqrt(3) - 1, but when minimizing f(t), the critical point is at t = √2 - 1.

This explains the discrepancy. Previously, I confused the functions to minimize. So let me correct this.

So the correct critical point is at t = √2 - 1. Now, let me compute f(t) at this value.

Let me set t = √2 - 1. Then:

1 + t = 1 + √2 - 1 = √2.

t(2 + t): Let me compute t = √2 - 1, so 2 + t = 2 + √2 - 1 = 1 + √2. Therefore, t(2 + t) = (√2 - 1)(1 + √2) = (√2)^2 - 1^2 = 2 - 1 = 1.

Therefore, f(t) = (1 + t)^2 / sqrt(t(2 + t)) = (√2)^2 / sqrt(1) = 2 / 1 = 2.

This matches the result from the Lagrange multipliers approach! So, the minimum value is 2, achieved at t = √2 - 1, which corresponds to x = z = a, y = ta = (√2 - 1)a. Let me compute a from the constraint.

From the constraint: a^4 * t(2 + t) = 1. Since t(2 + t) = 1, this implies a^4 = 1, so a = 1 (since a > 0). Therefore, x = z = 1, y = t * a = (√2 - 1) * 1 = √2 - 1, which is the same as the result from the Lagrange multipliers method.

Therefore, the minimum value of (x + y)(y + z) is 2.

This resolves the discrepancy. Earlier, I incorrectly derived the critical point for g(t) instead of f(t), leading to confusion. Now, with the correct derivative calculation, we find that the minimum occurs at t = √2 - 1, leading to the minimum value of 2.

Therefore, the minimum value is 2, achieved when x = z = 1 and y = √2 - 1.

To confirm, let's check with the earlier substitution approach: when t = √2 - 1, a = 1, so x = 1, y = √2 - 1, z = 1, then (x + y)(y + z) = (1 + √2 - 1)(√2 - 1 + 1) = √2 * √2 = 2, which matches.

Thus, after correcting the earlier mistakes in the substitution approach, the minimum value is indeed 2.

To ensure there are no other minima when x ≠ z, we can consider that the Lagrange multipliers method found this minimum, and due to the symmetry of the problem, it's likely that this is the global minimum.

Therefore, the minimum value of (x + y)(y + z) is 2.

**Final Answer**
\boxed{2}

To find the **minimum value** of the expression
$$
(x + y)(y + z)
$$
given the constraint
$$
xyz(x + y + z) = 1
$$
with $x, y, z > 0$, we proceed by leveraging symmetry and optimal substitution.

---

### Step 1: Use Symmetry and Substitution

Assume that $x = z$, based on the symmetry in the constraint and the expression to minimize. This helps reduce the number of variables and simplifies the analysis.

Let $x = z = a$, and $y = b$. Then the constraint becomes:
$$
a^2 b (2a + b) = 1
$$
The objective function becomes:
$$
(x + y)(y + z) = (a + b)^2
$$
So the problem reduces to minimizing:
$$
(a + b)^2 \quad \text{given} \quad a^2 b (2a + b) = 1
$$

---

### Step 2: Introduce a New Variable

Let $t = \frac{b}{a}$, so $b = ta$. Then:
$$
a^2 (ta)(2a + ta) = t a^3 (2 + t) = 1 \Rightarrow a^3 = \frac{1}{t(2 + t)}
$$

So:
$$
a = \left(\frac{1}{t(2 + t)}\right)^{1/3}
$$

Then:
$$
(a + b)^2 = (a(1 + t))^2 = \left(\left(\frac{1}{t(2 + t)}\right)^{1/3}(1 + t)\right)^2 = \frac{(1 + t)^2}{[t(2 + t)]^{2/3}}
$$

So we need to minimize:
$$
f(t) = \frac{(1 + t)^2}{[t(2 + t)]^{2/3}}
\quad \text{for} \quad t > 0
$$

---

### Step 3: Minimize Using Calculus

Take the derivative of $f(t)$, or simplify by analyzing critical points. A more direct and accurate route is to use **Lagrange Multipliers**.

---

### Step 4: Apply Lagrange Multipliers

Let the objective function be:
$$
f(x, y, z) = (x + y)(y + z)
$$
and constraint:
$$
g(x, y, z) = xyz(x + y + z) - 1 = 0
$$

Compute partial derivatives:
- $f_x = y + z$
- $f_y = x + 2y + z$
- $f_z = x + y$

- $g_x = yz(x + y + z) + xyz$
- $g_y = xz(x + y + z) + xyz$
- $g_z = xy(x + y + z) + xyz$

Set up the system:
$$
f_x = \lambda g_x, \quad f_y = \lambda g_y, \quad f_z = \lambda g_z
$$

Assume $x = z$, and consider the symmetry. After solving the system, we find that the minimum occurs when:
$$
z = x = 1, \quad y = \sqrt{2} - 1
$$

Substitute these into the expression:
$$
(x + y)(y + z) = (1 + (\sqrt{2} - 1))(\sqrt{2} - 1 + 1) = \sqrt{2} \cdot \sqrt{2} = 2
$$

---

### Step 5: Verify Constraint is Satisfied

Check the original constraint:
$$
xyz(x + y + z) = 1 \cdot (\sqrt{2} - 1) \cdot 1 \cdot (1 + (\sqrt{2} - 1) + 1) = (\sqrt{2} - 1)(1 + \sqrt{2}) = 2 - 1 = 1
$$
✅ Constraint is satisfied.

---

### Final Answer

$$
\boxed{2}
$$

Solution:
We can write $(x + y)(y + z)$ as $xz + y(x + y + z).$  By AM-GM,
\[xz + y(x + y + z) \ge 2 \sqrt{(xz)y(x + y + z)} = 2 \sqrt{xyz(x + y + z)} = 2.\]Equality holds when $xz = y(x + y + z) = 1$ and $xyz(x + y + z) = 1.$  For example, we can take $x = 1,$ $y = \sqrt{2} - 1,$ and $z = 1.$  Hence, the minimum value is $\boxed{2}.$