Explanation:
Okay, so I need to find the minimum value of the expression $x^4 + 4y^2 + 4z^4$ given that $xyz = 2$ where $x$, $y$, and $z$ are positive real numbers. Hmm, let me think about how to approach this.

First, since there's a constraint $xyz = 2$, maybe I can use methods like Lagrange multipliers? Or perhaps there's a way to apply the AM-GM inequality? I remember that AM-GM is useful for optimization problems with products and sums. Let me recall the AM-GM inequality: for non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean.

But the expression I need to minimize has different exponents: $x^4$, $y^2$, and $z^4$. Maybe I can adjust the terms so that they can be compared using AM-GM. Let me think.

Alternatively, maybe I can use substitution. Since $xyz = 2$, I can express one variable in terms of the other two. For example, $z = \frac{2}{xy}$. Then substitute this into the original expression to get everything in terms of $x$ and $y$. But then I would have to deal with a function of two variables, which might be more complicated. Maybe calculus would be needed here?

Alternatively, maybe I can use the method of Lagrange multipliers. Let me try that. Let me recall how that works. If I have a function $f(x, y, z)$ to minimize subject to a constraint $g(x, y, z) = 0$, then I can set up the equations $\nabla f = \lambda \nabla g$.

So here, my function to minimize is $f(x, y, z) = x^4 + 4y^2 + 4z^4$, and the constraint is $g(x, y, z) = xyz - 2 = 0$.

Calculating the gradients:

First, the gradient of $f$:

$\frac{\partial f}{\partial x} = 4x^3$

$\frac{\partial f}{\partial y} = 8y$

$\frac{\partial f}{\partial z} = 16z^3$

Gradient of $g$:

$\frac{\partial g}{\partial x} = yz$

$\frac{\partial g}{\partial y} = xz$

$\frac{\partial g}{\partial z} = xy$

So according to the method of Lagrange multipliers, we have:

$4x^3 = \lambda yz$ (1)

$8y = \lambda xz$ (2)

$16z^3 = \lambda xy$ (3)

And the constraint $xyz = 2$ (4)

So now I have four equations with four variables: $x$, $y$, $z$, and $\lambda$. Let me try to solve these equations.

First, maybe I can express $\lambda$ from equations (1), (2), and (3) and set them equal to each other.

From equation (1): $\lambda = \frac{4x^3}{yz}$

From equation (2): $\lambda = \frac{8y}{xz}$

Set them equal:

$\frac{4x^3}{yz} = \frac{8y}{xz}$

Multiply both sides by $yz \cdot xz$ to eliminate denominators:

$4x^3 \cdot xz = 8y \cdot yz$

Simplify:

Left side: $4x^4 z$

Right side: $8y^2 z$

Divide both sides by 4z:

$x^4 = 2y^2$

So, $x^4 = 2y^2$ → $y^2 = \frac{x^4}{2}$ → $y = \frac{x^2}{\sqrt{2}}$ since $y$ is positive.

Okay, so that's a relation between $x$ and $y$. Let me note that down: $y = \frac{x^2}{\sqrt{2}}$.

Now, let's take equations (2) and (3) to find another relation. Let me take equation (2): $\lambda = \frac{8y}{xz}$ and equation (3): $\lambda = \frac{16z^3}{xy}$. Set them equal:

$\frac{8y}{xz} = \frac{16z^3}{xy}$

Multiply both sides by $xz \cdot xy$:

$8y \cdot xy = 16z^3 \cdot xz$

Simplify:

Left side: $8x y^2$

Right side: $16x z^4$

Divide both sides by 8x:

$y^2 = 2 z^4$

So, $y^2 = 2 z^4$ → $y = \sqrt{2} z^2$ since $y$ is positive.

So now we have two expressions for $y$:

From earlier: $y = \frac{x^2}{\sqrt{2}}$

From this: $y = \sqrt{2} z^2$

Therefore, equate them:

$\frac{x^2}{\sqrt{2}} = \sqrt{2} z^2$

Multiply both sides by $\sqrt{2}$:

$x^2 = 2 z^2$ → $x = \sqrt{2} z$ since $x$ is positive.

So now, we have $x = \sqrt{2} z$ and $y = \sqrt{2} z^2$.

Now, let's use the constraint equation (4): $xyz = 2$.

Substitute $x$ and $y$ in terms of $z$:

$x = \sqrt{2} z$, $y = \sqrt{2} z^2$, so:

$xyz = (\sqrt{2} z)(\sqrt{2} z^2)(z) = (\sqrt{2} \cdot \sqrt{2}) \cdot z \cdot z^2 \cdot z = (2) \cdot z^{4} = 2 z^4 = 2$

Therefore, $2 z^4 = 2$ → $z^4 = 1$ → $z = 1$ since $z$ is positive.

So, $z = 1$. Then, from $x = \sqrt{2} z$, $x = \sqrt{2} \times 1 = \sqrt{2}$.

From $y = \sqrt{2} z^2$, $y = \sqrt{2} \times 1^2 = \sqrt{2}$.

So, the critical point is at $x = \sqrt{2}$, $y = \sqrt{2}$, $z = 1$.

Now, let me check if this gives a minimum. Since the function is positive and the constraint defines a compact set? Wait, actually, the domain is positive real numbers with $xyz = 2$, which is not compact. However, since the function tends to infinity as any variable tends to 0 or infinity, there should be a minimum. So, this critical point is likely the minimum.

Now, let me compute the value of the expression at this point:

$x^4 + 4y^2 + 4z^4 = (\sqrt{2})^4 + 4(\sqrt{2})^2 + 4(1)^4$.

Calculating each term:

$(\sqrt{2})^4 = (2)^{2} = 4$

$4(\sqrt{2})^2 = 4 \times 2 = 8$

$4(1)^4 = 4 \times 1 = 4$

Adding them up: 4 + 8 + 4 = 16.

So, the value is 16.

But wait, let me check if there's a possibility of a lower value. Maybe I made a mistake in the Lagrange multipliers? Let me verify the calculations again.

First, solving equations (1) and (2):

From (1): 4x³ = λ y z

From (2): 8y = λ x z

Dividing (1) by (2): (4x³)/(8y) = (λ y z)/(λ x z) → (x³)/(2y) = y/x → cross multiplying: x^4 = 2y². Correct.

Similarly, equations (2) and (3):

From (2): 8y = λ x z

From (3): 16 z³ = λ x y

Dividing (2) by (3): (8y)/(16 z³) = (λ x z)/(λ x y) → (y)/(2 z³) = z/y → cross multiplying: y² = 2 z^4. Correct.

Then, equating the two expressions for y:

From x^4 = 2 y² and y² = 2 z^4, so substituting y² from the second into the first: x^4 = 2*(2 z^4) = 4 z^4 → x^4 = 4 z^4 → x = √2 z. Correct.

Then, substituting into the constraint:

xyz = (√2 z)(√2 z²)(z) = √2 * √2 * z * z² * z = 2 * z^4 = 2 → z^4 = 1 → z = 1. Correct.

So, calculations seem correct. Then, substituting back gives x = √2, y = √2, z = 1. Then, the expression is 4 + 8 + 4 = 16.

But let me try another approach to verify. Maybe using AM-GM inequality. Let me think.

The original expression is $x^4 + 4y^2 + 4z^4$. Let me see if I can split the terms into parts that can be applied with AM-GM.

Note that AM-GM requires the terms to be positive, which they are. Let me think about how to split the terms.

Alternatively, maybe I can write the expression as the sum of terms whose product is a constant, given the constraint.

But the constraint is $xyz = 2$. Let me think.

Alternatively, maybe I can use substitution. Let me try to use substitution with the relations found earlier.

But since I already have a critical point, maybe I can check with specific numbers. For example, suppose I take x = √2, y = √2, z = 1. Then xyz = √2 * √2 * 1 = 2, which satisfies the constraint.

Alternatively, let me try another set of numbers. Suppose I take z = 1, then xyz = 2 implies xy = 2. Let me pick x = 2, y = 1, z = 1. Then the expression is 2^4 + 4*(1)^2 + 4*(1)^4 = 16 + 4 + 4 = 24, which is larger than 16.

Another example: take x = 1, y = 2, z = 1. Then xyz = 1*2*1 = 2. The expression is 1^4 + 4*(2)^2 + 4*(1)^4 = 1 + 16 + 4 = 21, still larger than 16.

What if I take x = sqrt(2), y = sqrt(2), z = 1? As before, gives 16.

What if I take z slightly different? Let me try z = 2. Then, from xyz = 2, if z = 2, then xy = 1. Let me take x = 1, y = 1, z = 2. Then the expression is 1^4 + 4*(1)^2 + 4*(2)^4 = 1 + 4 + 4*16 = 1 + 4 + 64 = 69, which is way bigger.

Alternatively, take z = 0.5. Then, xyz = 2 implies xy = 4. Let me take x = 2, y = 2, z = 0.5. Then the expression is 2^4 + 4*(2)^2 + 4*(0.5)^4 = 16 + 16 + 4*(1/16) = 16 + 16 + 0.25 = 32.25, which is still bigger than 16.

Alternatively, maybe take x, y, z closer to the critical point. Let me try x = sqrt(2) ≈ 1.414, y = sqrt(2) ≈ 1.414, z = 1. Let me check if perturbing slightly increases the value.

Suppose I take x = 1.5, then from x^4 = 2y², so y² = (1.5)^4 / 2. Let me compute that.

(1.5)^4 = (3/2)^4 = 81/16 ≈ 5.0625. Then y² ≈ 5.0625 / 2 ≈ 2.53125, so y ≈ 1.591. Then, from y² = 2 z^4, so z^4 = y² / 2 ≈ 2.53125 / 2 ≈ 1.2656, so z ≈ (1.2656)^{1/4}. Let me compute that.

First, sqrt(1.2656) ≈ 1.125, then sqrt(1.125) ≈ 1.06. So z ≈ 1.06. Then, check xyz: x = 1.5, y ≈ 1.591, z ≈ 1.06. Multiply them: 1.5 * 1.591 ≈ 2.3865, then * 1.06 ≈ 2.52. But the constraint is xyz = 2, so this is not satisfying. Therefore, need to adjust.

Alternatively, maybe this approach is getting too messy. Since the Lagrange multiplier method gave me a critical point, and testing other points gives higher values, maybe 16 is indeed the minimum.

But let me try another approach with AM-GM. Let me think about splitting the terms.

The expression is $x^4 + 4y^2 + 4z^4$. Let me note that 4y² can be thought of as 2y² + 2y². Similarly, 4z^4 can be thought of as z^4 + z^4 + z^4 + z^4. Maybe I can split the terms into more parts so that I can apply AM-GM with the constraint.

Alternatively, since the constraint is xyz = 2, maybe I need to relate the terms in the expression to the product xyz.

Let me think. Let me try to use weighted AM-GM.

Suppose I consider the terms in the expression. Let me think of the expression as a sum of terms, and try to find weights such that the product relates to the constraint.

Alternatively, maybe split the terms into multiple variables. For example, since we have x^4, y^2, z^4, maybe split x^4 into four terms of x^4/4 each, and 4y^2 into two terms of 2y^2 each, and 4z^4 into four terms of z^4 each. Wait, that might not be helpful.

Alternatively, think of the expression as x^4 + 2y^2 + 2y^2 + z^4 + z^4 + z^4 + z^4. So that's 1 + 2 + 4 terms? Wait, original expression is x^4 + 4y^2 + 4z^4. So splitting 4y^2 into two 2y^2 terms and 4z^4 into four z^4 terms. So total terms: 1 (x^4) + 2 (2y^2) + 4 (z^4) = 7 terms.

Then, applying AM-GM on these 7 terms.

The AM-GM inequality states that the arithmetic mean is greater than or equal to the geometric mean.

So, the arithmetic mean of these 7 terms is [x^4 + 2y^2 + 2y^2 + z^4 + z^4 + z^4 + z^4]/7.

Which is equal to [x^4 + 4y^2 + 4z^4]/7.

The geometric mean is (x^4 * (2y^2)^2 * (z^4)^4)^{1/7} ?

Wait, no. Wait, the terms are: one term of x^4, two terms of 2y^2, and four terms of z^4. Wait, no, if I split 4y^2 into two terms of 2y^2 each, and 4z^4 into four terms of z^4 each. So total terms: 1 + 2 + 4 = 7 terms.

So the geometric mean is (x^4 * (2y^2)^2 * (z^4)^4)^{1/7} ?

Wait, no. Wait, the geometric mean is the product of all terms raised to the power 1/7.

So the product is (x^4) * (2y^2) * (2y^2) * (z^4) * (z^4) * (z^4) * (z^4).

Which is x^4 * (2y^2)^2 * (z^4)^4.

Simplify:

x^4 * 4y^4 * z^{16}

So the geometric mean is [x^4 * 4y^4 * z^{16}]^{1/7}

Therefore, AM-GM says:

[x^4 + 4y^2 + 4z^4]/7 ≥ [x^4 * 4y^4 * z^{16}]^{1/7}

Therefore,

x^4 + 4y^2 + 4z^4 ≥ 7 * [x^4 * 4y^4 * z^{16}]^{1/7}

Hmm, but how does this relate to the constraint xyz = 2? Let me see.

Let me try to express the product inside the brackets in terms of xyz.

First, note that xyz = 2. Let me write the product:

x^4 * 4y^4 * z^{16} = 4 * x^4 y^4 z^{16}

But since xyz = 2, we can write x^4 y^4 z^{16} = (x y z)^4 * z^8 = (2)^4 * z^8 = 16 z^8.

Wait, let me check:

x^4 y^4 z^{16} = (x y z)^4 * z^{8} ?

Wait, (x y z)^4 = x^4 y^4 z^4. Therefore, x^4 y^4 z^{16} = (x y z)^4 * z^{12} ?

Wait, no. Let me think again.

Let me write x^4 y^4 z^{16} as (x y z)^4 * z^{8} ?

Wait, (x y z)^4 = x^4 y^4 z^4. Therefore, x^4 y^4 z^{16} = (x y z)^4 * z^{12} ?

Yes, because z^{16} = z^4 * z^{12}. Therefore, x^4 y^4 z^{16} = (x y z)^4 * z^{12} = (2)^4 * z^{12} = 16 z^{12}

Therefore, the product inside the brackets is 4 * 16 z^{12} = 64 z^{12}

Wait, no. Wait, original product was x^4 * 4y^4 * z^{16} = 4 * x^4 y^4 z^{16} = 4 * 16 z^{12} = 64 z^{12}

Therefore, the geometric mean is [64 z^{12}]^{1/7} = 64^{1/7} * z^{12/7}

But this seems to depend on z, which complicates things. Therefore, this approach may not be helpful because the geometric mean still depends on z.

Alternatively, maybe I need to adjust the way I split the terms. Maybe I need to split the terms in such a way that the product relates directly to the constraint.

Alternatively, maybe I can use Hölder's inequality? Not sure.

Alternatively, let me think again about the Lagrange multipliers result. Since that gave me a concrete answer, and testing other points gives higher values, maybe 16 is indeed the minimum.

But let me try another approach. Let me consider substituting variables.

From the earlier relations, we found that at the minimum, x = √2 z and y = √2 z². Therefore, substituting into the original expression:

x^4 + 4y² + 4z^4 = (√2 z)^4 + 4*(√2 z²)^2 + 4z^4 = (4 z^4) + 4*(2 z^4) + 4 z^4 = 4 z^4 + 8 z^4 + 4 z^4 = 16 z^4.

But from the constraint, we found that z = 1, so 16*(1)^4 = 16.

Alternatively, if I didn't know z, but expressed in terms of z, then the expression is 16 z^4. But since the constraint gives z^4 = 1, so 16*1 = 16.

But if I didn't use Lagrange multipliers, but instead thought of the expression as 16 z^4, then how would I know that z^4 = 1? Because from the constraint, we have xyz = 2.

But with x = √2 z and y = √2 z², then xyz = √2 z * √2 z² * z = 2 z^4 = 2 ⇒ z^4 = 1.

Therefore, the expression is 16 z^4 = 16*1 = 16.

So, this seems consistent.

Alternatively, maybe I can think of the expression as follows. Let me consider variables substitution. Let me set a = x^2, b = y, c = z^2. Then, the original expression becomes a² + 4b² + 4c².

But the constraint is xyz = 2. Let me express that in terms of a, b, c.

Since a = x² ⇒ x = √a, b = y, c = z² ⇒ z = √c. Therefore, the constraint is √a * b * √c = 2 ⇒ √(a c) * b = 2 ⇒ b √(a c) = 2.

So, the problem becomes: minimize a² + 4b² + 4c² subject to b √(a c) = 2.

Not sure if this helps. Maybe not directly.

Alternatively, maybe use AM-GM on a², 4b², 4c². Let me think.

But the constraint is b √(a c) = 2. Maybe I need to relate a, b, c.

Alternatively, maybe set variables such that the constraint becomes a product. Let me think.

Alternatively, maybe use substitution. Let me set t = √(a c). Then, the constraint is b t = 2 ⇒ b = 2 / t.

But then, since t = √(a c), then t² = a c.

So, the expression to minimize is a² + 4*(4 / t²) + 4c². Wait, no. Wait, original expression is a² + 4b² + 4c². Since b = 2 / t, then 4b² = 4*(4 / t²) = 16 / t².

So, the expression becomes a² + 16 / t² + 4c². But since t² = a c, so c = t² / a. Therefore, substitute c into the expression:

a² + 16 / t² + 4*(t² / a)^2 = a² + 16 / t² + 4 t^4 / a².

Now, this is a function of a and t. Maybe set u = a², v = t². Then, the expression becomes u + 16 / v + 4 v² / u.

But this seems complicated. Maybe set u = v? Not sure.

Alternatively, for fixed v, minimize over u. Let me think. For fixed v, the expression is u + 4 v² / u + 16 / v. The minimum over u is when derivative with respect to u is zero.

Let me compute derivative: d/du [u + 4 v² / u] = 1 - 4 v² / u². Setting to zero: 1 - 4 v² / u² = 0 ⇒ u² = 4 v² ⇒ u = 2 v.

Therefore, the minimum for fixed v is when u = 2v. Therefore, substituting back, the expression becomes 2v + 4 v² / (2v) + 16 / v = 2v + 2v + 16 / v = 4v + 16 / v.

Now, we need to minimize 4v + 16 / v over v > 0.

This is straightforward. Take derivative: 4 - 16 / v². Set to zero: 4 - 16 / v² = 0 ⇒ 4 = 16 / v² ⇒ v² = 4 ⇒ v = 2.

Therefore, minimum is 4*2 + 16 / 2 = 8 + 8 = 16.

So, the minimum value is 16.

Therefore, this approach also leads to 16.

Therefore, this confirms the previous result.

So, regardless of the method, the minimum value is 16.

Therefore, I think the answer is 16.

But just to make sure, let me check if there's another way. Maybe using substitution with the original variables.

From the Lagrange multipliers, we found that at minimum, x = √2 z, y = √2 z². Therefore, substituting into the original expression gives 16 z^4, and from the constraint, z^4 = 1. Therefore, 16*1 = 16.

Alternatively, if I didn't use Lagrange multipliers, but thought of the expression as x^4 + 4y^2 + 4z^4 and tried to use AM-GM on the terms.

Let me think. Let me consider the terms x^4, 4y^2, and 4z^4. Maybe split them into multiple terms so that the number of terms corresponds to the exponents.

For example, since x^4 is a single term, 4y^2 can be considered as two terms of 2y^2 each, and 4z^4 can be considered as four terms of z^4 each. So total terms: 1 + 2 + 4 = 7 terms.

Then, applying AM-GM on these 7 terms:

(x^4 + 2y^2 + 2y^2 + z^4 + z^4 + z^4 + z^4)/7 ≥ (x^4 * (2y^2)^2 * (z^4)^4)^{1/7}

Which simplifies to:

(x^4 + 4y^2 + 4z^4)/7 ≥ [x^4 * 4y^4 * z^{16}]^{1/7}

But as before, this leads to:

x^4 + 4y^2 + 4z^4 ≥ 7 * [x^4 * 4y^4 * z^{16}]^{1/7}

But how to relate this to the constraint xyz = 2?

Let me express x^4 * y^4 * z^{16} in terms of (xyz)^something.

Note that xyz = 2. Let me write x^4 * y^4 * z^{16} = (x y z)^4 * z^{8} = (2)^4 * z^8 = 16 z^8.

Therefore, the expression becomes:

x^4 + 4y^2 + 4z^4 ≥ 7 * [4 * 16 z^8]^{1/7} = 7 * [64 z^8]^{1/7} = 7 * 64^{1/7} * z^{8/7}

But this still depends on z. However, if we can find a relation between z and the constraint, but since we have xyz = 2, and we have relations from the AM-GM equality condition.

Equality in AM-GM occurs when all terms are equal. So, for the AM-GM above, equality holds when:

x^4 = 2y^2 = z^4.

Wait, because the terms are x^4, 2y^2, 2y^2, z^4, z^4, z^4, z^4. So equality holds when all these terms are equal.

Therefore, x^4 = 2y^2 = z^4.

So, let me set x^4 = 2y^2 = z^4 = k.

Then, x^4 = k ⇒ x = k^{1/4}

Similarly, 2y^2 = k ⇒ y^2 = k/2 ⇒ y = sqrt(k/2)

Similarly, z^4 = k ⇒ z = k^{1/4}

Therefore, x = z, and y = sqrt(k/2).

Now, using the constraint xyz = 2:

x * y * z = x * y * x = x² * y = 2.

Since x = z, and x = k^{1/4}, y = sqrt(k/2).

So, substituting:

(k^{1/4})² * sqrt(k/2) = k^{1/2} * (k/2)^{1/2} = [k^{1/2} * k^{1/2} / 2^{1/2}] = [k / 2^{1/2}] = 2

Therefore:

k / √2 = 2 ⇒ k = 2√2

Therefore, k = 2√2.

Therefore, x^4 = k = 2√2 ⇒ x = (2√2)^{1/4}

Similarly, z^4 = k = 2√2 ⇒ z = same as x.

And y^2 = k / 2 = (2√2)/2 = √2 ⇒ y = (√2)^{1/2} = 2^{1/4}

Now, compute the original expression x^4 + 4y^2 + 4z^4.

Since x^4 = k = 2√2, z^4 = k = 2√2, and 4y^2 = 4*(√2) = 4√2.

Therefore, total expression is 2√2 + 4√2 + 2√2 = 8√2 ≈ 11.31... Wait, but this contradicts the previous result of 16.

Wait, this can't be. There must be a mistake here.

Wait, let me check the calculation again.

If equality holds when all terms are equal, so x^4 = 2y^2 = z^4.

Let me denote x^4 = z^4 = k, so x = z.

Then, 2y^2 = k ⇒ y^2 = k/2.

Then, the constraint is xyz = 2.

Since x = z, then x * y * z = x^2 * y = 2.

But x^4 = k ⇒ x^2 = sqrt(k).

Therefore, sqrt(k) * y = 2.

But y^2 = k/2 ⇒ y = sqrt(k/2).

Therefore, sqrt(k) * sqrt(k/2) = 2 ⇒ sqrt(k * k / 2) = 2 ⇒ sqrt(k² / 2) = 2 ⇒ k / sqrt(2) = 2 ⇒ k = 2 * sqrt(2).

Therefore, k = 2√2.

Then, the original expression is x^4 + 4y^2 + 4z^4 = k + 4*(k/2) + k = k + 2k + k = 4k = 4*(2√2) = 8√2 ≈ 11.31.

But this contradicts the previous result of 16.

Wait, so which one is correct?

But earlier, using Lagrange multipliers, I found that the minimum is 16, but here, using AM-GM with splitting into 7 terms, I get 8√2 ≈ 11.31.

But this suggests that there is a contradiction. Therefore, I must have made a mistake in one of the approaches.

Wait, let me check the AM-GM approach again.

When I split the expression into 7 terms: x^4, 2y^2, 2y^2, z^4, z^4, z^4, z^4.

Then, AM-GM says that the arithmetic mean is greater or equal to the geometric mean.

But the arithmetic mean is [x^4 + 4y^2 + 4z^4]/7.

The geometric mean is [x^4 * (2y^2)^2 * (z^4)^4]^{1/7} = [x^4 * 4y^4 * z^{16}]^{1/7}.

But then, I said that x^4 * 4y^4 * z^{16} = 4 * x^4 y^4 z^{16} = 4 * (xyz)^4 * z^8.

Wait, but xyz = 2, so (xyz)^4 = 16. Therefore, 4 * 16 * z^8 = 64 z^8.

Therefore, geometric mean is [64 z^8]^{1/7} = 64^{1/7} * z^{8/7}.

But then, the AM-GM inequality gives:

[x^4 + 4y^2 + 4z^4]/7 ≥ 64^{1/7} * z^{8/7}

But this still depends on z. However, if we assume equality condition, which requires all terms to be equal, i.e., x^4 = 2y^2 = z^4.

But then, with that, we found that the expression equals 8√2 ≈ 11.31, but according to the Lagrange multipliers, the minimum is 16.

This inconsistency suggests that there is a mistake in one of the methods.

Wait, let me check with the values obtained from the AM-GM equality condition.

If k = 2√2, then x^4 = 2√2, so x = (2√2)^{1/4} = (2^{3/2})^{1/4} = 2^{3/8} ≈ 2^{0.375} ≈ 1.296.

Similarly, z = x ≈ 1.296.

y^2 = k / 2 = (2√2)/2 = √2 ⇒ y = (√2)^{1/2} = 2^{1/4} ≈ 1.189.

Check the constraint xyz = 2:

x * y * z ≈ 1.296 * 1.189 * 1.296 ≈ Let me compute:

First, 1.296 * 1.296 ≈ 1.68, then * 1.189 ≈ 1.68 * 1.189 ≈ 2.0.

Yes, that's approximately 2. So the constraint is satisfied.

Now, compute the original expression x^4 + 4y^2 + 4z^4.

x^4 = 2√2 ≈ 2.828,

4y^2 = 4*(√2) ≈ 5.656,

4z^4 = 4*(2√2) ≈ 11.313.

Adding them up: 2.828 + 5.656 + 11.313 ≈ 19.797.

Wait, but earlier I thought it was 8√2 ≈ 11.31. Wait, what's wrong here?

Wait, earlier I said that the expression is 4k, where k = 2√2. But that was incorrect.

Wait, original expression is x^4 + 4y^2 + 4z^4.

Since x^4 = k, z^4 = k, and 4y^2 = 4*(k/2) = 2k.

Therefore, total expression is k + 2k + k = 4k.

But k = 2√2, so 4k = 8√2 ≈ 11.31.

But when I calculated numerically, I got approximately 19.797.

Wait, there's a contradiction here. Where is the mistake?

Wait, let me recalculate with k = 2√2.

x^4 = k = 2√2 ≈ 2.828.

z^4 = k = 2√2 ≈ 2.828.

4y^2 = 4*(k/2) = 2k ≈ 2*(2.828) ≈ 5.656.

So total expression is 2.828 + 5.656 + 2.828 ≈ 11.312, which is 8√2 ≈ 11.31.

But when I calculated numerically earlier, I thought x ≈ 1.296, y ≈ 1.189, z ≈ 1.296.

Let me compute x^4: (1.296)^4.

First, 1.296^2 ≈ 1.68, then squared: 1.68^2 ≈ 2.822, which is ≈ 2.828, which is 2√2.

Similarly, z^4 is same.

4y^2: y ≈ 1.189, so y^2 ≈ 1.414, which is √2, so 4y^2 ≈ 4*1.414 ≈ 5.656.

So total expression is 2.828 + 5.656 + 2.828 ≈ 11.312.

But earlier, when I thought I calculated it as ≈ 19.797, I must have made a mistake.

So, the value is indeed approximately 11.31, which is less than 16.

But this contradicts the Lagrange multipliers result.

So, there must be a mistake in one of the methods.

Wait, let me check the Lagrange multipliers again.

Earlier, I found that at the critical point, x = √2, y = √2, z = 1.

Let me check if this satisfies the equality condition of the AM-GM approach.

In the AM-GM approach, equality holds when x^4 = 2y^2 = z^4.

Let me check for x = √2, y = √2, z = 1.

x^4 = (√2)^4 = 4.

2y^2 = 2*(√2)^2 = 2*2 = 4.

z^4 = 1^4 = 1.

So, x^4 = 2y^2 = 4, but z^4 = 1 ≠ 4. Therefore, equality condition is not satisfied.

Therefore, the critical point found by Lagrange multipliers does not satisfy the equality condition of the AM-GM approach.

Therefore, there must be a mistake in assuming that splitting into 7 terms and applying AM-GM gives the minimum.

But why?

Wait, because when I split the terms, I might have introduced an incorrect relation.

Alternatively, maybe the AM-GM approach here is not tight enough, or the way of splitting terms is not appropriate.

Alternatively, maybe there are multiple critical points, and the one found by Lagrange multipliers is a local minimum, but there is another one with lower value?

But how?

Wait, let me check the value of the expression at the AM-GM equality point.

It is approximately 11.31, which is less than 16.

But according to the Lagrange multipliers, the critical point gives 16.

This suggests that there is a mistake in one of the methods.

Wait, let me check the Lagrange multipliers again.

From the Lagrange equations, we had:

From equations (1) and (2): x^4 = 2y^2

From equations (2) and (3): y^2 = 2z^4

Therefore, combining these: x^4 = 2*(2z^4) = 4z^4 ⇒ x^2 = 2z^2 ⇒ x = √2 z

Then, from the constraint xyz = 2, substituting x = √2 z and y = sqrt(2) z², we get:

(√2 z) * (sqrt(2) z²) * z = (√2 * sqrt(2)) * z * z² * z = 2 * z^4 = 2 ⇒ z^4 = 1 ⇒ z = 1.

Therefore, x = √2, y = sqrt(2), z = 1.

But according to the AM-GM approach, there is another critical point where x^4 = 2y^2 = z^4, which gives a lower value.

But why didn't the Lagrange multipliers find this?

Wait, maybe because when I set up the Lagrange multipliers, I assumed that the gradient of f is proportional to the gradient of g, but maybe there are multiple critical points?

Wait, let me check if the point from the AM-GM approach satisfies the Lagrange conditions.

Let me take the point where x^4 = 2y^2 = z^4 = k.

Let me denote k = 2√2 as before.

So, x^4 = k, y^2 = k/2, z^4 = k.

So, x = k^{1/4}, y = (k/2)^{1/2}, z = k^{1/4}.

Let me compute the partial derivatives at this point.

First, compute the gradient of f:

∇f = (4x³, 8y, 16z³)

Gradient of g: (yz, xz, xy)

According to Lagrange multipliers, there exists λ such that:

4x³ = λ yz

8y = λ xz

16z³ = λ xy

Let me check if these equations are satisfied.

First, compute λ from first equation: λ = 4x³ / (yz)

From second equation: λ = 8y / (xz)

Set equal: 4x³ / (yz) = 8y / (xz) ⇒ 4x³ * xz = 8y * yz ⇒ 4x^4 z = 8y² z ⇒ 4x^4 = 8y² ⇒ x^4 = 2y², which is satisfied.

Similarly, from second and third equations:

From second equation: λ = 8y / (xz)

From third equation: λ = 16z³ / (xy)

Set equal: 8y / (xz) = 16z³ / (xy) ⇒ 8y * xy = 16z³ * xz ⇒ 8x y² = 16x z^4 ⇒ 8y² = 16z^4 ⇒ y² = 2z^4, which is also satisfied.

Therefore, this point does satisfy the Lagrange conditions.

But earlier, when I solved the Lagrange equations, I found x = √2 z, y = √2 z², and z = 1.

But here, with x^4 = 2y² = z^4, we have x = z, and y^2 = x^4 / 2.

Wait, so if x = z, then from x^4 = z^4, which is true.

But earlier, from the Lagrange equations, we had x = √2 z, which would imply that x = z only if z = 0, which is not possible.

Wait, this suggests that there are two different critical points?

Wait, no. Wait, in the first approach, solving the Lagrange equations gave us x = √2 z, y = √2 z², and from the constraint, z = 1.

But in the second approach, assuming equality in AM-GM, we have x^4 = 2y² = z^4, which gives x = z, y = sqrt(x^4 / 2).

So, these are two different sets of relations.

Wait, but how can there be two different critical points?

Wait, let me check if both sets of relations are compatible.

From the first set: x = √2 z, y = √2 z².

From the second set: x = z, y = sqrt(x^4 / 2).

So, if we assume both, then x = √2 z and x = z ⇒ z = 0, which is not allowed.

Therefore, these are two different critical points?

But how?

Wait, no. Wait, the Lagrange equations should have only one solution?

Wait, let me check.

From the Lagrange equations, we had:

From equations (1) and (2): x^4 = 2y²

From equations (2) and (3): y² = 2z^4

Therefore, combining these gives x^4 = 4z^4 ⇒ x² = 2z² ⇒ x = √2 z

Then, from the constraint, we found z = 1.

But in the AM-GM approach, we assumed equality in AM-GM, which gave us x^4 = 2y² = z^4, which is different from x^4 = 4z^4.

Wait, so there is a contradiction here.

Wait, if x^4 = 2y² and y² = 2z^4, then x^4 = 2*(2z^4) = 4z^4, so x^4 = 4z^4, which implies x = √2 z.

But in the AM-GM approach, we assumed x^4 = 2y² = z^4, which would mean that 2y² = z^4 and x^4 = z^4, so x^4 = z^4 = 2y².

But from the Lagrange equations, we have x^4 = 4z^4, so unless z = 0, which it isn't, these are different.

Therefore, the AM-GM approach must have an error.

Wait, why did the AM-GM approach give a different result?

Because when I split the expression into 7 terms, I might have introduced an incorrect assumption.

Let me check the AM-GM step again.

I split the expression into x^4, 2y^2, 2y^2, z^4, z^4, z^4, z^4.

But the original expression is x^4 + 4y^2 + 4z^4.

So, splitting 4y^2 into two terms of 2y^2 each, and 4z^4 into four terms of z^4 each.

But then, the AM-GM inequality says that the arithmetic mean is greater than or equal to the geometric mean.

But the problem is that the terms are not all positive? No, they are positive.

But why does this give a different result?

Wait, perhaps because the AM-GM approach here is not tight enough, or the way of splitting terms is not appropriate.

Alternatively, maybe there is a mistake in assuming that the minimum occurs at the equality condition of AM-GM.

But why?

Wait, let me check the value of the expression at the AM-GM equality point.

As calculated earlier, it is approximately 11.31, which is less than 16.

But according to the Lagrange multipliers, the minimum is 16.

This suggests that there is a contradiction.

Wait, let me check if the AM-GM equality point satisfies the constraint.

Yes, as checked earlier, xyz ≈ 2.

But then, why does the Lagrange multiplier method give a higher value?

Wait, perhaps I made a mistake in solving the Lagrange equations?

Let me re-examine the Lagrange equations.

From the Lagrange equations:

From (1): 4x³ = λ yz

From (2): 8y = λ xz

From (3): 16z³ = λ xy

From (1) and (2): 4x³ / (yz) = 8y / (xz) ⇒ 4x^4 z = 8y² z ⇒ 4x^4 = 8y² ⇒ x^4 = 2y²

From (2) and (3): 8y / (xz) = 16z³ / (xy) ⇒ 8y * xy = 16z³ * xz ⇒ 8x y² = 16x z^4 ⇒ 8y² = 16z^4 ⇒ y² = 2z^4

So, x^4 = 2y² = 2*(2z^4) = 4z^4 ⇒ x^4 = 4z^4 ⇒ x = √2 z

Then, from the constraint xyz = 2:

x = √2 z, y = sqrt(2z^4) = sqrt(2) z²

So, xyz = √2 z * sqrt(2) z² * z = (√2 * sqrt(2)) * z^4 = 2 * z^4 = 2 ⇒ z^4 = 1 ⇒ z = 1

So, x = √2, y = √2, z = 1.

Now, let me compute the value of the expression at this point:

x^4 + 4y^2 + 4z^4 = (√2)^4 + 4*(√2)^2 + 4*(1)^4 = 4 + 4*2 + 4 = 4 + 8 + 4 = 16.

But earlier, with the AM-GM approach, I found a point where the expression is approximately 11.31.

But how is this possible?

Wait, let me check if this point satisfies the Lagrange conditions.

Let me take the point from the AM-GM approach: x^4 = 2y² = z^4 = k = 2√2.

So, x = (2√2)^{1/4} = (2^{3/2})^{1/4} = 2^{3/8} ≈ 1.296

y = sqrt(k/2) = sqrt(2√2 / 2) = sqrt(√2) = 2^{1/4} ≈ 1.189

z = (2√2)^{1/4} ≈ 1.296

Now, let me check if these values satisfy the Lagrange equations.

Compute ∇f = (4x³, 8y, 16z³)

Compute ∇g = (yz, xz, xy)

Let me compute λ from the first equation: λ = 4x³ / (yz)

Compute 4x³ / (yz):

x ≈ 1.296, y ≈ 1.189, z ≈ 1.296

Compute x³ ≈ (1.296)^3 ≈ 2.177

4x³ ≈ 8.708

yz ≈ 1.189 * 1.296 ≈ 1.541

So λ ≈ 8.708 / 1.541 ≈ 5.65

From the second equation: λ = 8y / (xz)

Compute 8y / (xz):

8 * 1.189 / (1.296 * 1.296) ≈ 9.512 / 1.68 ≈ 5.66

Which is approximately equal to 5.65, so this is consistent.

From the third equation: λ = 16z³ / (xy)

Compute 16z³ / (xy):

z³ ≈ (1.296)^3 ≈ 2.177

16 * 2.177 ≈ 34.83

xy ≈ 1.296 * 1.189 ≈ 1.541

So λ ≈ 34.83 / 1.541 ≈ 22.6

Wait, this is not equal to 5.65!

Wait, this is a problem.

So, according to the first two equations, λ ≈ 5.65, but according to the third equation, λ ≈ 22.6.

This inconsistency suggests that this point does not satisfy all three Lagrange equations, which means that my assumption of equality in AM-GM was incorrect.

Wait, but earlier, I thought that equality in AM-GM requires all terms to be equal. Let me check if all terms are equal.

The terms are: x^4, 2y^2, 2y^2, z^4, z^4, z^4, z^4.

At the AM-GM point, x^4 = k = 2√2, 2y^2 = 2*(k/2) = k = 2√2, and z^4 = k = 2√2.

So, all terms are equal to k = 2√2.

Therefore, the AM-GM equality condition is satisfied.

But why does it not satisfy the Lagrange equations?

Wait, this suggests that there is a mistake in the calculation of λ from the third equation.

Let me recalculate λ from the third equation.

Given x ≈ 1.296, y ≈ 1.189, z ≈ 1.296

Compute 16z³ / (xy):

First, z³ = (1.296)^3 ≈ 2.177

16 * 2.177 ≈ 34.83

xy = 1.296 * 1.189 ≈ 1.541

So, 34.83 / 1.541 ≈ 22.6

But from the first equation, λ ≈ 5.65.

This inconsistency suggests that this point does not satisfy all three Lagrange equations, which is a contradiction.

But why?

Wait, because when I applied AM-GM, I split the expression into 7 terms, but the Lagrange equations are derived from the original function and constraint.

This suggests that there is a mistake in the AM-GM approach.

Wait, let me check the AM-GM calculation again.

The AM-GM inequality says that the arithmetic mean is greater than or equal to the geometric mean.

But in this case, the arithmetic mean is [x^4 + 4y^2 + 4z^4]/7, and the geometric mean is [x^4 * (2y^2)^2 * (z^4)^4]^{1/7} = [x^4 * 4y^4 * z^{16}]^{1/7}.

But if all terms are equal, then x^4 = 2y^2 = z^4 = k, so x^4 = k, 2y^2 = k ⇒ y^2 = k/2, z^4 = k.

Then, the product is x^4 * 4y^4 * z^{16} = k * 4*(k/2)^2 * k^4 = k * 4*(k²/4) * k^4 = k * k² * k^4 = k^7.

Therefore, the geometric mean is [k^7]^{1/7} = k.

Therefore, AM-GM says that [x^4 + 4y^2 + 4z^4]/7 ≥ k ⇒ x^4 + 4y^2 + 4z^4 ≥ 7k.

But in this case, x^4 + 4y^2 + 4z^4 = k + 2k + k = 4k.

So, 4k ≥ 7k ⇒ 4 ≥ 7, which is false.

Wait, this is a contradiction!

This means that my AM-GM application was incorrect.

Wait, what's the issue here?

Ah! I think I made a mistake in the number of terms.

When I split the expression into 7 terms, I said: x^4, 2y^2, 2y^2, z^4, z^4, z^4, z^4.

But the original expression is x^4 + 4y^2 + 4z^4.

So, splitting 4y^2 into two terms of 2y^2 each, and 4z^4 into four terms of z^4 each.

But the number of terms is 1 + 2 + 4 = 7.

But the AM-GM inequality says that the arithmetic mean of these 7 terms is greater than or equal to the geometric mean.

But if all terms are equal, then the arithmetic mean is equal to the geometric mean.

But in this case, if all terms are equal to k, then the arithmetic mean is k, and the geometric mean is also k.

But the arithmetic mean is [x^4 + 4y^2 + 4z^4]/7 = [k + 2k + k]/7 = 4k/7.

Wait, no! Wait, if all terms are equal to k, then the sum is 7k, so the arithmetic mean is 7k / 7 = k.

But the original expression is x^4 + 4y^2 + 4z^4 = k + 2k + k = 4k.

Wait, this is a contradiction.

Wait, no! Wait, the original expression is x^4 + 4y^2 + 4z^4.

But when I split it into 7 terms, it's x^4 + 2y^2 + 2y^2 + z^4 + z^4 + z^4 + z^4.

If all these terms are equal to k, then the sum is 7k, so the original expression is 7k.

But earlier, I said that x^4 + 4y^2 + 4z^4 = 4k.

This is the mistake!

Ah! Here is the error.

If I split the expression into 7 terms, each equal to k, then the original expression is 7k.

But earlier, I thought that x^4 + 4y^2 + 4z^4 = k + 2k + k = 4k, which is incorrect.

Actually, if x^4 = k, 2y^2 = k, and z^4 = k, then the original expression is x^4 + 4y^2 + 4z^4 = k + 4*(k/2) + 4k = k + 2k + 4k = 7k.

Ah! So, the original expression is 7k, not 4k.

Therefore, the AM-GM inequality says that 7k ≥ 7k, which is equality.

Therefore, the minimum value is 7k, and from the constraint, we can find k.

Wait, let me re-express this.

If all terms are equal to k, then the original expression is 7k.

But from the constraint, we have xyz = 2.

Let me find k in terms of the constraint.

From the AM-GM equality conditions: x^4 = 2y^2 = z^4 = k.

So, x^4 = k, z^4 = k ⇒ x = z.

Also, 2y^2 = k ⇒ y^2 = k/2.

From the constraint: xyz = x * y * z = x^2 * y = 2.

Since x^4 = k, x^2 = sqrt(k).

And y = sqrt(k/2).

So, sqrt(k) * sqrt(k/2) = 2 ⇒ sqrt(k * k / 2) = 2 ⇒ sqrt(k² / 2) = 2 ⇒ k / sqrt(2) = 2 ⇒ k = 2√2.

Therefore, the original expression is 7k = 7 * 2√2 = 14√2 ≈ 19.798.

Wait, but earlier, when I calculated the expression at this point, I got approximately 11.31, but that was a mistake.

Actually, the expression is 7k = 7 * 2√2 ≈ 19.798, which is higher than 16.

So, this resolves the contradiction.

Earlier, I mistakenly thought that the expression was 4k, but it is actually 7k.

Therefore, the AM-GM approach gives a higher value than the Lagrange multiplier method, which suggests that the minimum is indeed 16.

This shows the importance of careful calculation.

Therefore, the correct minimum value is 16, as found by the Lagrange multipliers.

Thus, the minimum value of the expression is 16, achieved at x = √2, y = √2, z = 1.

**Final Answer**
\boxed{16}

To find the minimum value of the expression
$$
x^4 + 4y^2 + 4z^4
$$
under the constraint
$$
xyz = 2,
$$
where $ x, y, z > 0 $, we can use the method of **Lagrange multipliers**.

---

### Step 1: Define the Objective and Constraint

Let
$$
f(x, y, z) = x^4 + 4y^2 + 4z^4
$$
be the function to minimize, and
$$
g(x, y, z) = xyz - 2 = 0
$$
be the constraint.

---

### Step 2: Compute Gradients

We compute the gradients of $ f $ and $ g $:

- $ \nabla f = (4x^3, 8y, 16z^3) $
- $ \nabla g = (yz, xz, xy) $

According to the method of Lagrange multipliers, we set:
$$
\nabla f = \lambda \nabla g
$$
This gives the system of equations:
$$
\begin{cases}
4x^3 = \lambda yz \\
8y = \lambda xz \\
16z^3 = \lambda xy \\
xyz = 2
\end{cases}
$$

---

### Step 3: Solve the System

From the first two equations:
$$
\frac{4x^3}{yz} = \frac{8y}{xz} \Rightarrow 4x^4 z = 8y^2 z \Rightarrow x^4 = 2y^2
$$

From the second and third equations:
$$
\frac{8y}{xz} = \frac{16z^3}{xy} \Rightarrow 8x y^2 = 16x z^4 \Rightarrow y^2 = 2z^4
$$

Now substitute $ y^2 = 2z^4 $ into $ x^4 = 2y^2 $:
$$
x^4 = 2(2z^4) = 4z^4 \Rightarrow x = \sqrt{2} z
$$

Now substitute $ x = \sqrt{2} z $ and $ y^2 = 2z^4 $ into the constraint $ xyz = 2 $:
$$
(\sqrt{2} z)(\sqrt{2} z^2)(z) = 2 \Rightarrow 2z^4 = 2 \Rightarrow z^4 = 1 \Rightarrow z = 1
$$

Then:
- $ x = \sqrt{2} $
- $ y = \sqrt{2} $

---

### Step 4: Evaluate the Expression

Now substitute $ x = \sqrt{2}, y = \sqrt{2}, z = 1 $ into the original expression:
$$
x^4 + 4y^2 + 4z^4 = (\sqrt{2})^4 + 4(\sqrt{2})^2 + 4(1)^4 = 4 + 8 + 4 = 16
$$

---

### Step 5: Verify Minimality

Since the function $ f(x, y, z) $ is smooth and the constraint defines a compact set (in the positive reals), and the function tends to infinity as any variable approaches 0 or infinity, the critical point found is indeed the global minimum.

---

### Final Answer

$$
\boxed{16}
$$

[Assistant]
Explanation:
Okay, so I need to find the minimum value of the expression $x^4 + 4y^2 + 4z^4$ given that $xyz = 2$ where $x$, $y$, and $z$ are positive real numbers. Hmm, let me think about how to approach this.

First, since there's a constraint $xyz = 2$, maybe I can use methods like Lagrange multipliers? Or perhaps there's a way to apply the AM-GM inequality? I remember that AM-GM is useful for optimization problems with products and sums. Let me recall the AM-GM inequality: for non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean.

But the expression I need to minimize has different exponents: $x^4$, $y^2$, and $z^4$. Maybe I can adjust the terms so that they can be compared using AM-GM. Let me think.

Alternatively, maybe I can use substitution. Since $xyz = 2$, I can express one variable in terms of the other two. For example, $z = \frac{2}{xy}$. Then substitute this into the original expression to get everything in terms of $x$ and $y$. But then I would have to deal with a function of two variables, which might be more complicated. Maybe calculus would be needed here?

Alternatively, maybe I can use the method of Lagrange multipliers. Let me try that. Let me recall how that works. If I have a function $f(x, y, z)$ to minimize subject to a constraint $g(x, y, z) = 0$, then I can set up the equations $\nabla f = \lambda \nabla g$.

So here, my function to minimize is $f(x, y, z) = x^4 + 4y^2 + 4z^4$, and the constraint is $g(x, y, z) = xyz - 2 = 0$.

Calculating the gradients:

First, the gradient of $f$:

$\frac{\partial f}{\partial x} = 4x^3$

$\frac{\partial f}{\partial y} = 8y$

$\frac{\partial f}{\partial z} = 16z^3$

Gradient of $g$:

$\frac{\partial g}{\partial x} = yz$

$\frac{\partial g}{\partial y} = xz$

$\frac{\partial g}{\partial z} = xy$

So according to the method of Lagrange multipliers, we have:

$4x^3 = \lambda yz$ (1)

$8y = \lambda xz$ (2)

$16z^3 = \lambda xy$ (3)

And the constraint $xyz = 2$ (4)

So now I have four equations with four variables: $x$, $y$, $z$, and $\lambda$. Let me try to solve these equations.

First, maybe I can express $\lambda$ from equations (1), (2), and (3) and set them equal to each other.

From equation (1): $\lambda = \frac{4x^3}{yz}$

From equation (2): $\lambda = \frac{8y}{xz}$

Set them equal:

$\frac{4x^3}{yz} = \frac{8y}{xz}$

Multiply both sides by $yz \cdot xz$ to eliminate denominators:

$4x^3 \cdot xz = 8y \cdot yz$

Simplify:

Left side: $4x^4 z$

Right side: $8y^2 z$

Divide both sides by 4z:

$x^4 = 2y^2$

So, $x^4 = 2y^2$ → $y^2 = \frac{x^4}{2}$ → $y = \frac{x^2}{\sqrt{2}}$ since $y$ is positive.

Okay, so that's a relation between $x$ and $y$. Let me note that down: $y = \frac{x^2}{\sqrt{2}}$.

Now, let's take equations (2) and (3) to find another relation. Let me take equation (2): $\lambda = \frac{8y}{xz}$ and equation (3): $\lambda = \frac{16z^3}{xy}$. Set them equal:

$\frac{8y}{xz} = \frac{16z^3}{xy}$

Multiply both sides by $xz \cdot xy$:

$8y \cdot xy = 16z^3 \cdot xz$

Simplify:

Left side: $8x y^2$

Right side: $16x z^4$

Divide both sides by 8x:

$y^2 = 2 z^4$

So, $y^2 = 2 z^4$ → $y = \sqrt{2} z^2$ since $y$ is positive.

So now we have two expressions for $y$:

From earlier: $y = \frac{x^2}{\sqrt{2}}$

From this: $y = \sqrt{2} z^2$

Therefore, equate them:

$\frac{x^2}{\sqrt{2}} = \sqrt{2} z^2$

Multiply both sides by $\sqrt{2}$:

$x^2 = 2 z^2$ → $x = \sqrt{2} z$ since $x$ is positive.

So now, we have $x = \sqrt{2} z$ and $y = \sqrt{2} z^2$.

Now, let's use the constraint equation (4): $xyz = 2$.

Substitute $x$ and $y$ in terms of $z$:

$x = \sqrt{2} z$, $y = \sqrt{2} z^2$, so:

$xyz = (\sqrt{2} z)(\sqrt{2} z^2)(z) = (\sqrt{2} \cdot \sqrt{2}) \cdot z \cdot z^2 \cdot z = (2) \cdot z^{4} = 2 z^4 = 2$

Therefore, $2 z^4 = 2$ → $z^4 = 1$ → $z = 1$ since $z$ is positive.

So, $z = 1$. Then, from $x = \sqrt{2} z$, $x = \sqrt{2} \times 1 = \sqrt{2}$.

From $y = \sqrt{2} z^2$, $y = \sqrt{2} \times 1^2 = \sqrt{2}$.

So, the critical point is at $x = \sqrt{2}$, $y = \sqrt{2}$, $z = 1$.

Now, let me check if this gives a minimum. Since the function is positive and the constraint defines a compact set? Wait, actually, the domain is positive real numbers with $xyz = 2$, which is not compact. However, since the function tends to infinity as any variable tends to 0 or infinity, there should be a minimum. So, this critical point is likely the minimum.

Now, let me compute the value of the expression at this point:

$x^4 + 4y^2 + 4z^4 = (\sqrt{2})^4 + 4(\sqrt{2})^2 + 4(1)^4$.

Calculating each term:

$(\sqrt{2})^4 = (2)^{2} = 4$

$4(\sqrt{2})^2 = 4 \times 2 = 8$

$4(1)^4 = 4 \times 1 = 4$

Adding them up: 4 + 8 + 4 = 16.

So, the value is 16.

But wait, let me check if there's a possibility of a lower value. Maybe I made a mistake in the Lagrange multipliers? Let me verify the calculations again.

First, solving equations (1) and (2):

From (1): 4x³ = λ y z

From (2): 8y = λ x z

Dividing (1) by (2): (4x³)/(8y) = (λ y z)/(λ x z) → (x³)/(2y) = y/x → cross multiplying: x^4 = 2y². Correct.

Similarly, equations (2) and (3):

From (2): 8y = λ x z

From (3): 16 z³ = λ x y

Dividing (2) by (3): (8y)/(16 z³) = (λ x z)/(λ x y) → (y)/(2 z³) = z/y → cross multiplying: y² = 2 z^4. Correct.

Then, equating the two expressions for y:

From x^4 = 2 y² and y² = 2 z^4, so substituting y² from the second into the first: x^4 = 2*(2 z^4) = 4 z^4 → x^4 = 4 z^4 → x = √2 z. Correct.

Then, substituting into the constraint:

xyz = (√2 z)(√2 z²)(z) = √2 * √2 * z * z² * z = 2 * z^4 = 2 → z^4 = 1 → z = 1. Correct.

So, calculations seem correct. Then, substituting back gives x = √2, y = √2, z = 1. Then, the expression is 4 + 8 + 4 = 16.

But let me try another approach to verify. Maybe using AM-GM inequality. Let me think.

The original expression is $x^4 + 4y^2 + 4z^4$. Let me see if I can split the terms into parts that can be applied with AM-GM.

Note that AM-GM requires the terms to be positive, which they are. Let me think about how to split the terms.

Alternatively, maybe I can write the expression as the sum of terms whose product is a constant, given the constraint.

But the constraint is $xyz = 2$. Let me think.

Alternatively, maybe I can use substitution. Let me try to use substitution with the relations found earlier.

But since I already have a critical point, maybe I can check with specific numbers. For example, suppose I take x = √2, y = √2, z = 1. Then xyz = √2 * √2 * 1 = 2, which satisfies the constraint.

Alternatively, let me try another set of numbers. Suppose I take z = 1, then xyz = 2 implies xy = 2. Let me pick x = 2, y = 1, z = 1. Then the expression is 2^4 + 4*(1)^2 + 4*(1)^4 = 16 + 4 + 4 = 24, which is larger than 16.

Another example: take x = 1, y = 2, z = 1. Then xyz = 1*2*1 = 2. The expression is 1^4 + 4*(2)^2 + 4*(1)^4 = 1 + 16 + 4 = 21, still larger than 16.

What if I take x = sqrt(2), y = sqrt(2), z = 1? As before, gives 16.

What if I take z slightly different? Let me try z = 2. Then, from xyz = 2, if z = 2, then xy = 1. Let me take x = 1, y = 1, z = 2. Then the expression is 1^4 + 4*(1)^2 + 4*(2)^4 = 1 + 4 + 4*16 = 1 + 4 + 64 = 69, which is way bigger.

Alternatively, take z = 0.5. Then, xyz = 2 implies xy = 4. Let me take x = 2, y = 2, z = 0.5. Then the expression is 2^4 + 4*(2)^2 + 4*(0.5)^4 = 16 + 16 + 4*(1/16) = 16 + 16 + 0.25 = 32.25, which is still bigger than 16.

Alternatively, maybe take x, y, z closer to the critical point. Let me try x = sqrt(2) ≈ 1.414, y = sqrt(2) ≈ 1.414, z = 1. Let me check if perturbing slightly increases the value.

Suppose I take x = 1.5, then from x^4 = 2y², so y² = (1.5)^4 / 2. Let me compute that.

(1.5)^4 = (3/2)^4 = 81/16 ≈ 5.0625. Then y² ≈ 5.0625 / 2 ≈ 2.53125, so y ≈ 1.591. Then, from y² = 2 z^4, so z^4 = y² / 2 ≈ 2.53125 / 2 ≈ 1.2656, so z ≈ (1.2656)^{1/4}. Let me compute that.

First, sqrt(1.2656) ≈ 1.125, then sqrt(1.125) ≈ 1.06. So z ≈ 1.06. Then, check xyz: x = 1.5, y ≈ 1.591, z ≈ 1.06. Multiply them: 1.5 * 1.591 ≈ 2.3865, then * 1.06 ≈ 2.52. But the constraint is xyz = 2, so this is not satisfying. Therefore, need to adjust.

Alternatively, maybe this approach is getting too messy. Since the Lagrange multiplier method gave me a critical point, and testing other points gives higher values, maybe 16 is indeed the minimum.

But let me try another approach with AM-GM. Let me think about splitting the terms.

The expression is $x^4 + 4y^2 + 4z^4$. Let me note that 4y² can be thought of as 2y² + 2y². Similarly, 4z^4 can be thought of as z^4 + z^4 + z^4 + z^4. Maybe I can split the terms into more parts so that I can apply AM-GM with the constraint.

Alternatively, since the constraint is xyz = 2, maybe I need to relate the terms in the expression to the product xyz.

Let me think. Let me try to use weighted AM-GM.

Suppose I consider the terms in the expression. Let me think of the expression as a sum of terms, and try to find weights such that the product relates to the constraint.

Alternatively, maybe split the terms into multiple variables. For example, since we have x^4, y^2, z^4, maybe split x^4 into four terms of x^4/4 each, and 4y^2 into two terms of 2y^2 each, and 4z^4 into four terms of z^4 each. Wait, that might not be helpful.

Alternatively, think of the expression as x^4 + 2y^2 + 2y^2 + z^4 + z^4 + z^4 + z^4. So that's 1 + 2 + 4 terms? Wait, original expression is x^4 + 4y^2 + 4z^4. So splitting 4y^2 into two 2y^2 terms and 4z^4 into four z^4 terms. So total terms: 1 (x^4) + 2 (2y^2) + 4 (z^4) = 7 terms.

Then, applying AM-GM on these 7 terms.

The AM-GM inequality states that the arithmetic mean is greater than or equal to the geometric mean.

So, the arithmetic mean of these 7 terms is [x^4 + 2y^2 + 2y^2 + z^4 + z^4 + z^4 + z^4]/7.

Which is equal to [x^4 + 4y^2 + 4z^4]/7.

The geometric mean is (x^4 * (2y^2)^2 * (z^4)^4)^{1/7} ?

Wait, no. Wait, the terms are: one term of x^4, two terms of 2y^2, and four terms of z^4. Wait, no, if I split 4y^2 into two terms of 2y^2 each, and 4z^4 into four terms of z^4 each. So total terms: 1 + 2 + 4 = 7 terms.

So the geometric mean is (x^4 * (2y^2)^2 * (z^4)^4)^{1/7} ?

Wait, no. Wait, the geometric mean is the product of all terms raised to the power 1/7.

So the product is (x^4) * (2y^2) * (2y^2) * (z^4) * (z^4) * (z^4) * (z^4).

Which is x^4 * (2y^2)^2 * (z^4)^4.

Simplify:

x^4 * 4y^4 * z^{16}

So the geometric mean is [x^4 * 4y^4 * z^{16}]^{1/7}

Therefore, AM-GM says:

[x^4 + 4y^2 + 4z^4]/7 ≥ [x^4 * 4y^4 * z^{16}]^{1/7}

Therefore,

x^4 + 4y^2 + 4z^4 ≥ 7 * [x^4 * 4y^4 * z^{16}]^{1/7}

Hmm, but how does this relate to the constraint xyz = 2? Let me see.

Let me try to express the product inside the brackets in terms of xyz.

First, note that xyz = 2. Let me write the product:

x^4 * 4y^4 * z^{16} = 4 * x^4 y^4 z^{16}

But since xyz = 2, we can write x^4 y^4 z^{16} = (x y z)^4 * z^8 = (2)^4 * z^8 = 16 z^8.

Wait, let me check:

x^4 y^4 z^{16} = (x y z)^4 * z^{8} ?

Wait, (x y z)^4 = x^4 y^4 z^4. Therefore, x^4 y^4 z^{16} = (x y z)^4 * z^{12} ?

Wait, no. Let me think again.

Let me write x^4 y^4 z^{16} as (x y z)^4 * z^{8} ?

Wait, (x y z)^4 = x^4 y^4 z^4. Therefore, x^4 y^4 z^{16} = (x y z)^4 * z^{12} ?

Yes, because z^{16} = z^4 * z^{12}. Therefore, x^4 y^4 z^{16} = (x y z)^4 * z^{12} = (2)^4 * z^{12} = 16 z^{12}

Therefore, the product inside the brackets is 4 * 16 z^{12} = 64 z^{12}

Wait, no. Wait, original product was x^4 * 4y^4 * z^{16} = 4 * x^4 y^4 z^{16} = 4 * 16 z^{12} = 64 z^{12}

Therefore, the geometric mean is [64 z^{12}]^{1/7} = 64^{1/7} * z^{12/7}

But this seems to depend on z, which complicates things. Therefore, this approach may not be helpful because the geometric mean still depends on z.

Alternatively, maybe I need to adjust the way I split the terms. Maybe I need to split the terms in such a way that the product relates directly to the constraint.

Alternatively, maybe I can use Hölder's inequality? Not sure.

Alternatively, let me think again about the Lagrange multipliers result. Since that gave me a concrete answer, and testing other points gives higher values, maybe 16 is indeed the minimum.

But let me try another approach. Let me consider substituting variables.

From the earlier relations, we found that at the minimum, x = √2 z and y = √2 z². Therefore, substituting into the original expression:

x^4 + 4y² + 4z^4 = (√2 z)^4 + 4*(√2 z²)^2 + 4z^4 = (4 z^4) + 4*(2 z^4) + 4 z^4 = 4 z^4 + 8 z^4 + 4 z^4 = 16 z^4.

But from the constraint, we found that z = 1, so 16*(1)^4 = 16.

Alternatively, if I didn't know z, but expressed in terms of z, then the expression is 16 z^4. But since the constraint gives z^4 = 1, so 16*1 = 16.

But if I didn't use Lagrange multipliers, but instead thought of the expression as 16 z^4, then how would I know that z^4 = 1? Because from the constraint, we have xyz = 2.

But with x = √2 z and y = √2 z², then xyz = √2 z * √2 z² * z = 2 z^4 = 2 ⇒ z^4 = 1.

Therefore, the expression is 16 z^4 = 16*1 = 16.

So, this seems consistent.

Alternatively, maybe I can think of the expression as follows. Let me consider variables substitution. Let me set a = x^2, b = y, c = z^2. Then, the original expression becomes a² + 4b² + 4c².

But the constraint is xyz = 2. Let me express that in terms of a, b, c.

Since a = x² ⇒ x = √a, b = y, c = z² ⇒ z = √c. Therefore, the constraint is √a * b * √c = 2 ⇒ √(a c) * b = 2 ⇒ b √(a c) = 2.

So, the problem becomes: minimize a² + 4b² + 4c² subject to b √(a c) = 2.

Not sure if this helps. Maybe not directly.

Alternatively, maybe use AM-GM on a², 4b², 4c². Let me think.

But the constraint is b √(a c) = 2. Maybe I need to relate a, b, c.

Alternatively, maybe set variables such that the constraint becomes a product. Let me think.

Alternatively, maybe use substitution. Let me set t = √(a c). Then, the constraint is b t = 2 ⇒ b = 2 / t.

But then, since t = √(a c), then t² = a c.

So, the expression to minimize is a² + 4*(4 / t²) + 4c². Wait, no. Wait, original expression is a² + 4b² + 4c². Since b = 2 / t, then 4b² = 4*(4 / t²) = 16 / t².

So, the expression becomes a² + 16 / t² + 4c². But since t² = a c, so c = t² / a. Therefore, substitute c into the expression:

a² + 16 / t² + 4*(t² / a)^2 = a² + 16 / t² + 4 t^4 / a².

Now, this is a function of a and t. Maybe set u = a², v = t². Then, the expression becomes u + 16 / v + 4 v² / u.

But this seems complicated. Maybe set u = v? Not sure.

Alternatively, for fixed v, minimize over u. Let me think. For fixed v, the expression is u + 4 v² / u + 16 / v. The minimum over u is when derivative with respect to u is zero.

Let me compute derivative: d/du [u + 4 v² / u] = 1 - 4 v² / u². Setting to zero: 1 - 4 v² / u² = 0 ⇒ u² = 4 v² ⇒ u = 2 v.

Therefore, the minimum for fixed v is when u = 2v. Therefore, substituting back, the expression becomes 2v + 4 v² / (2v) + 16 / v = 2v + 2v + 16 / v = 4v + 16 / v.

Now, we need to minimize 4v + 16 / v over v > 0.

This is straightforward. Take derivative: 4 - 16 / v². Set to zero: 4 - 16 / v² = 0 ⇒ 4 = 16 / v² ⇒ v² = 4 ⇒ v = 2.

Therefore, minimum is 4*2 + 16 / 2 = 8 + 8 = 16.

So, the minimum value is 16.

Therefore, this approach also leads to 16.

Therefore, this confirms the previous result.

So, regardless of the method, the minimum value is 16.

Therefore, I think the answer is 16.

But just to make sure, let me check if there's another way. Maybe using substitution with the original variables.

From the Lagrange multipliers, we found that at minimum, x = √2 z, y = √2 z². Therefore, substituting into the original expression gives 16 z^4, and from the constraint, z^4 = 1. Therefore, 16*1 = 16.

Alternatively, if I didn't use Lagrange multipliers, but thought of the expression as x^4 + 4y^2 + 4z^4 and tried to use AM-GM on the terms.

Let me think. Let me consider the terms x^4, 4y^2, and 4z^4. Maybe split them into multiple terms so that the number of terms corresponds to the exponents.

For example, since x^4 is a single term, 4y^2 can be considered as two terms of 2y^2 each, and 4z^4 can be considered as four terms of z^4 each. So total terms: 1 + 2 + 4 = 7 terms.

Then, applying AM-GM on these 7 terms:

(x^4 + 2y^2 + 2y^2 + z^4 + z^4 + z^4 + z^4)/7 ≥ (x^4 * (2y^2)^2 * (z^4)^4)^{1/7}

Which simplifies to:

(x^4 + 4y^2 + 4z^4)/7 ≥ [x^4 * 4y^4 * z^{16}]^{1/7}

But as before, this leads to:

x^4 + 4y^2 + 4z^4 ≥ 7 * [x^4 * 4y^4 * z^{16}]^{1/7}

But how to relate this to the constraint xyz = 2?

Let me express x^4 * y^4 * z^{16} in terms of (xyz)^something.

Note that xyz = 2. Let me write x^4 * y^4 * z^{16} = (x y z)^4 * z^{8} = (2)^4 * z^8 = 16 z^8.

Therefore, the expression becomes:

x^4 + 4y^2 + 4z^4 ≥ 7 * [4 * 16 z^8]^{1/7} = 7 * [64 z^8]^{1/7} = 7 * 64^{1/7} * z^{8/7}

But this still depends on z. However, if we can find a relation between z and the constraint, but since we have xyz = 2, and we have relations from the AM-GM equality condition.

Equality in AM-GM occurs when all terms are equal. So, for the AM-GM above, equality holds when:

x^4 = 2y^2 = z^4.

Wait, because the terms are x^4, 2y^2, 2y^2, z^4, z^4, z^4, z^4. So equality holds when all these terms are equal.

Therefore, x^4 = 2y^2 = z^4.

So, let me set x^4 = 2y^2 = z^4 = k.

Then, x^4 = k ⇒ x = k^{1/4}

Similarly, 2y^2 = k ⇒ y^2 = k/2 ⇒ y = sqrt(k/2)

Similarly, z^4 = k ⇒ z = k^{1/4}

Therefore, x = z, and y = sqrt(k/2).

Now, using the constraint xyz = 2:

x * y * z = x * y * x = x² * y = 2.

Since x = z, and x = k^{1/4}, y = sqrt(k/2).

So, substituting:

(k^{1/4})² * sqrt(k/2) = k^{1/2} * (k/2)^{1/2} = [k^{1/2} * k^{1/2} / 2^{1/2}] = [k / 2^{1/2}] = 2

Therefore:

k / √2 = 2 ⇒ k = 2√2

Therefore, k = 2√2.

Therefore, x^4 = k = 2√2 ⇒ x = (2√2)^{1/4}

Similarly, z^4 = k = 2√2 ⇒ z = same as x.

And y^2 = k / 2 = (2√2)/2 = √2 ⇒ y = (√2)^{1/2} = 2^{1/4}

Now, compute the original expression x^4 + 4y^2 + 4z^4.

Since x^4 = k = 2√2, z^4 = k = 2√2, and 4y^2 = 4*(√2) = 4√2.

Therefore, total expression is 2√2 + 4√2 + 2√2 = 8√2 ≈ 11.31... Wait, but this contradicts the previous result of 16.

Wait, this can't be. There must be a mistake here.

Wait, let me check the calculation again.

If equality holds when all terms are equal, so x^4 = 2y^2 = z^4.

Let me denote x^4 = z^4 = k, so x = z.

Then, 2y^2 = k ⇒ y^2 = k/2.

Then, the constraint is xyz = 2.

Since x = z, then x * y * z = x^2 * y = 2.

But x^4 = k ⇒ x^2 = sqrt(k).

Therefore, sqrt(k) * y = 2.

But y^2 = k/2 ⇒ y = sqrt(k/2).

Therefore, sqrt(k) * sqrt(k/2) = 2 ⇒ sqrt(k * k / 2) = 2 ⇒ sqrt(k² / 2) = 2 ⇒ k / sqrt(2) = 2 ⇒ k = 2 * sqrt(2).

Therefore, k = 2√2.

Then, the original expression is x^4 + 4y^2 + 4z^4 = k + 4*(k/2) + k = k + 2k + k = 4k = 4*(2√2) = 8√2 ≈ 11.31.

But this contradicts the previous result of 16.

Wait, so which one is correct?

But earlier, using Lagrange multipliers, I found that the minimum is 16, but here, using AM-GM with splitting into 7 terms, I get 8√2 ≈ 11.31.

But this suggests that there is a contradiction. Therefore, I must have made a mistake in one of the approaches.

Wait, let me check the AM-GM approach again.

When I split the expression into 7 terms: x^4, 2y^2, 2y^2, z^4, z^4, z^4, z^4.

Then, AM-GM says that the arithmetic mean is greater or equal to the geometric mean.

But the arithmetic mean is [x^4 + 4y^2 + 4z^4]/7.

The geometric mean is [x^4 * (2y^2)^2 * (z^4)^4]^{1/7} = [x^4 * 4y^4 * z^{16}]^{1/7}.

But then, I said that x^4 * 4y^4 * z^{16} = 4 * x^4 y^4 z^{16} = 4 * (xyz)^4 * z^8.

Wait, but xyz = 2, so (xyz)^4 = 16. Therefore, 4 * 16 * z^8 = 64 z^8.

Therefore, geometric mean is [64 z^8]^{1/7} = 64^{1/7} * z^{8/7}.

But then, the AM-GM inequality gives:

[x^4 + 4y^2 + 4z^4]/7 ≥ 64^{1/7} * z^{8/7}

But this still depends on z. However, if we assume equality condition, which requires all terms to be equal, i.e., x^4 = 2y^2 = z^4.

But then, with that, we found that the expression equals 8√2 ≈ 11.31, but according to the Lagrange multipliers, the minimum is 16.

This inconsistency suggests that there is a mistake in one of the methods.

Wait, let me check with the values obtained from the AM-GM equality condition.

If k = 2√2, then x^4 = 2√2, so x = (2√2)^{1/4} = (2^{3/2})^{1/4} = 2^{3/8} ≈ 2^{0.375} ≈ 1.296.

Similarly, z = x ≈ 1.296.

y^2 = k / 2 = (2√2)/2 = √2 ⇒ y = (√2)^{1/2} = 2^{1/4} ≈ 1.189.

Check the constraint xyz = 2:

x * y * z ≈ 1.296 * 1.189 * 1.296 ≈ Let me compute:

First, 1.296 * 1.296 ≈ 1.68, then * 1.189 ≈ 1.68 * 1.189 ≈ 2.0.

Yes, that's approximately 2. So the constraint is satisfied.

Now, compute the original expression x^4 + 4y^2 + 4z^4.

x^4 = 2√2 ≈ 2.828,

4y^2 = 4*(√2) ≈ 5.656,

4z^4 = 4*(2√2) ≈ 11.313.

Adding them up: 2.828 + 5.656 + 11.313 ≈ 19.797.

Wait, but earlier I thought it was 8√2 ≈ 11.31. Wait, what's wrong here?

Wait, earlier I said that the expression is 4k, where k = 2√2. But that was incorrect.

Wait, original expression is x^4 + 4y^2 + 4z^4.

Since x^4 = k, z^4 = k, and 4y^2 = 4*(k/2) = 2k.

Therefore, total expression is k + 2k + k = 4k.

But k = 2√2, so 4k = 8√2 ≈ 11.31.

But when I calculated numerically, I got approximately 19.797.

Wait, there's a contradiction here. Where is the mistake?

Wait, let me recalculate with k = 2√2.

x^4 = k = 2√2 ≈ 2.828.

z^4 = k = 2√2 ≈ 2.828.

4y^2 = 4*(k/2) = 2k ≈ 2*(2.828) ≈ 5.656.

So total expression is 2.828 + 5.656 + 2.828 ≈ 11.312, which is 8√2 ≈ 11.31.

But when I calculated numerically earlier, I thought x ≈ 1.296, y ≈ 1.189, z ≈ 1.296.

Let me compute x^4: (1.296)^4.

First, 1.296^2 ≈ 1.68, then squared: 1.68^2 ≈ 2.822, which is ≈ 2.828, which is 2√2.

Similarly, z^4 is same.

4y^2: y ≈ 1.189, so y^2 ≈ 1.414, which is √2, so 4y^2 ≈ 4*1.414 ≈ 5.656.

So total expression is 2.828 + 5.656 + 2.828 ≈ 11.312.

But earlier, when I thought I calculated it as ≈ 19.797, I must have made a mistake.

So, the value is indeed approximately 11.31, which is less than 16.

But this contradicts the Lagrange multipliers result.

So, there must be a mistake in one of the methods.

Wait, let me check the Lagrange multipliers again.

Earlier, I found that at the critical point, x = √2, y = √2, z = 1.

Let me check if this satisfies the equality condition of the AM-GM approach.

In the AM-GM approach, equality holds when x^4 = 2y^2 = z^4.

Let me check for x = √2, y = √2, z = 1.

x^4 = (√2)^4 = 4.

2y^2 = 2*(√2)^2 = 2*2 = 4.

z^4 = 1^4 = 1.

So, x^4 = 2y^2 = 4, but z^4 = 1 ≠ 4. Therefore, equality condition is not satisfied.

Therefore, the critical point found by Lagrange multipliers does not satisfy the equality condition of the AM-GM approach.

Therefore, there must be a mistake in assuming that splitting into 7 terms and applying AM-GM gives the minimum.

But why?

Wait, because when I split the terms, I might have introduced an incorrect relation.

Alternatively, maybe the AM-GM approach here is not tight enough, or the way of splitting terms is not appropriate.

Alternatively, maybe there are multiple critical points, and the one found by Lagrange multipliers is a local minimum, but there is another one with lower value?

But how?

Wait, let me check the value of the expression at the AM-GM equality point.

It is approximately 11.31, which is less than 16.

But according to the Lagrange multipliers, the critical point gives 16.

This suggests that there is a mistake in one of the methods.

Wait, let me check the Lagrange multipliers again.

From the Lagrange equations, we had:

From equations (1) and (2): x^4 = 2y^2

From equations (2) and (3): y^2 = 2z^4

Therefore, combining these: x^4 = 2*(2z^4) = 4z^4 ⇒ x^2 = 2z^2 ⇒ x = √2 z

Then, from the constraint xyz = 2, substituting x = √2 z and y = sqrt(2) z², we get:

(√2 z) * (sqrt(2) z²) * z = (√2 * sqrt(2)) * z * z² * z = 2 * z^4 = 2 ⇒ z^4 = 1 ⇒ z = 1.

Therefore, x = √2, y = sqrt(2), z = 1.

But according to the AM-GM approach, there is another critical point where x^4 = 2y^2 = z^4, which gives a lower value.

But why didn't the Lagrange multipliers find this?

Wait, maybe because when I set up the Lagrange multipliers, I assumed that the gradient of f is proportional to the gradient of g, but maybe there are multiple critical points?

Wait, let me check if the point from the AM-GM approach satisfies the Lagrange conditions.

Let me take the point where x^4 = 2y^2 = z^4 = k.

Let me denote k = 2√2 as before.

So, x^4 = k, y^2 = k/2, z^4 = k.

So, x = k^{1/4}, y = (k/2)^{1/2}, z = k^{1/4}.

Let me compute the partial derivatives at this point.

First, compute the gradient of f:

∇f = (4x³, 8y, 16z³)

Gradient of g: (yz, xz, xy)

According to Lagrange multipliers, there exists λ such that:

4x³ = λ yz

8y = λ xz

16z³ = λ xy

Let me check if these equations are satisfied.

First, compute λ from first equation: λ = 4x³ / (yz)

From second equation: λ = 8y / (xz)

Set equal: 4x³ / (yz) = 8y / (xz) ⇒ 4x³ * xz = 8y * yz ⇒ 4x^4 z = 8y² z ⇒ 4x^4 = 8y² ⇒ x^4 = 2y², which is satisfied.

Similarly, from second and third equations:

From second equation: λ = 8y / (xz)

From third equation: λ = 16z³ / (xy)

Set equal: 8y / (xz) = 16z³ / (xy) ⇒ 8y * xy = 16z³ * xz ⇒ 8x y² = 16x z^4 ⇒ 8y² = 16z^4 ⇒ y² = 2z^4, which is also satisfied.

Therefore, this point does satisfy the Lagrange conditions.

But earlier, when I solved the Lagrange equations, I found x = √2 z, y = √2 z², and z = 1.

But here, with x^4 = 2y² = z^4, we have x = z, and y^2 = x^4 / 2.

Wait, so if x = z, then from x^4 = z^4, which is true.

But earlier, from the Lagrange equations, we had x = √2 z, which would imply that x = z only if z = 0, which is not possible.

Wait, this suggests that there are two different critical points?

Wait, no. Wait, in the first approach, solving the Lagrange equations gave us x = √2 z, y = √2 z², and from the constraint, z = 1.

But in the second approach, assuming equality in AM-GM, we have x^4 = 2y² = z^4, which gives x = z, y = sqrt(x^4 / 2).

So, these are two different sets of relations.

Wait, but how can there be two different critical points?

Wait, let me check if both sets of relations are compatible.

From the first set: x = √2 z, y = √2 z².

From the second set: x = z, y = sqrt(x^4 / 2).

So, if we assume both, then x = √2 z and x = z ⇒ z = 0, which is not allowed.

Therefore, these are two different critical points?

But how?

Wait, no. Wait, the Lagrange equations should have only one solution?

Wait, let me check.

From the Lagrange equations, we had:

From equations (1) and (2): x^4 = 2y²

From equations (2) and (3): y² = 2z^4

Therefore, combining these gives x^4 = 4z^4 ⇒ x² = 2z² ⇒ x = √2 z

Then, from the constraint, we found z = 1.

But in the AM-GM approach, we assumed equality in AM-GM, which gave us x^4 = 2y² = z^4, which is different from x^4 = 4z^4.

Wait, so there is a contradiction here.

Wait, if x^4 = 2y² and y² = 2z^4, then x^4 = 2*(2z^4) = 4z^4, so x^4 = 4z^4, which implies x = √2 z.

But in the AM-GM approach, we assumed x^4 = 2y² = z^4, which would mean that 2y² = z^4 and x^4 = z^4, so x^4 = z^4 = 2y².

But from the Lagrange equations, we have x^4 = 4z^4, so unless z = 0, which it isn't, these are different.

Therefore, the AM-GM approach must have an error.

Wait, why did the AM-GM approach give a different result?

Because when I split the expression into 7 terms, I might have introduced an incorrect assumption.

Let me check the AM-GM step again.

I split the expression into x^4, 2y^2, 2y^2, z^4, z^4, z^4, z^4.

But the original expression is x^4 + 4y^2 + 4z^4.

So, splitting 4y^2 into two terms of 2y^2 each, and 4z^4 into four terms of z^4 each.

But then, the AM-GM inequality says that the arithmetic mean is greater than or equal to the geometric mean.

But the problem is that the terms are not all positive? No, they are positive.

But why does this give a different result?

Wait, perhaps because the AM-GM approach here is not tight enough, or the way of splitting terms is not appropriate.

Alternatively, maybe there is a mistake in assuming that the minimum occurs at the equality condition of AM-GM.

But why?

Wait, let me check the value of the expression at the AM-GM equality point.

As calculated earlier, it is approximately 11.31, which is less than 16.

But according to the Lagrange multipliers, the minimum is 16.

This suggests that there is a contradiction.

Wait, let me check if the AM-GM equality point satisfies the constraint.

Yes, as checked earlier, xyz ≈ 2.

But then, why does the Lagrange multiplier method give a higher value?

Wait, perhaps I made a mistake in solving the Lagrange equations?

Let me re-examine the Lagrange equations.

From the Lagrange equations:

From (1): 4x³ = λ yz

From (2): 8y = λ xz

From (3): 16z³ = λ xy

From (1) and (2): 4x³ / (yz) = 8y / (xz) ⇒ 4x^4 z = 8y² z ⇒ 4x^4 = 8y² ⇒ x^4 = 2y²

From (2) and (3): 8y / (xz) = 16z³ / (xy) ⇒ 8y * xy = 16z³ * xz ⇒ 8x y² = 16x z^4 ⇒ 8y² = 16z^4 ⇒ y² = 2z^4

So, x^4 = 2y² = 2*(2z^4) = 4z^4 ⇒ x^4 = 4z^4 ⇒ x = √2 z

Then, from the constraint xyz = 2:

x = √2 z, y = sqrt(2z^4) = sqrt(2) z²

So, xyz = √2 z * sqrt(2) z² * z = (√2 * sqrt(2)) * z^4 = 2 * z^4 = 2 ⇒ z^4 = 1 ⇒ z = 1

So, x = √2, y = √2, z = 1.

Now, let me compute the value of the expression at this point:

x^4 + 4y^2 + 4z^4 = (√2)^4 + 4*(√2)^2 + 4*(1)^4 = 4 + 4*2 + 4 = 4 + 8 + 4 = 16.

But earlier, with the AM-GM approach, I found a point where the expression is approximately 11.31.

But how is this possible?

Wait, let me check if this point satisfies the Lagrange conditions.

Let me take the point from the AM-GM approach: x^4 = 2y² = z^4 = k = 2√2.

So, x = (2√2)^{1/4} = (2^{3/2})^{1/4} = 2^{3/8} ≈ 1.296

y = sqrt(k/2) = sqrt(2√2 / 2) = sqrt(√2) = 2^{1/4} ≈ 1.189

z = (2√2)^{1/4} ≈ 1.296

Now, let me check if these values satisfy the Lagrange equations.

Compute ∇f = (4x³, 8y, 16z³)

Compute ∇g = (yz, xz, xy)

Let me compute λ from the first equation: λ = 4x³ / (yz)

Compute 4x³ / (yz):

x ≈ 1.296, y ≈ 1.189, z ≈ 1.296

Compute x³ ≈ (1.296)^3 ≈ 2.177

4x³ ≈ 8.708

yz ≈ 1.189 * 1.296 ≈ 1.541

So λ ≈ 8.708 / 1.541 ≈ 5.65

From the second equation: λ = 8y / (xz)

Compute 8y / (xz):

8 * 1.189 / (1.296 * 1.296) ≈ 9.512 / 1.68 ≈ 5.66

Which is approximately equal to 5.65, so this is consistent.

From the third equation: λ = 16z³ / (xy)

Compute 16z³ / (xy):

z³ ≈ (1.296)^3 ≈ 2.177

16 * 2.177 ≈ 34.83

xy ≈ 1.296 * 1.189 ≈ 1.541

So λ ≈ 34.83 / 1.541 ≈ 22.6

Wait, this is not equal to 5.65!

Wait, this is a problem.

So, according to the first two equations, λ ≈ 5.65, but according to the third equation, λ ≈ 22.6.

This inconsistency suggests that this point does not satisfy all three Lagrange equations, which means that my assumption of equality in AM-GM was incorrect.

Wait, but earlier, I thought that equality in AM-GM requires all terms to be equal. Let me check if all terms are equal.

The terms are: x^4, 2y^2, 2y^2, z^4, z^4, z^4, z^4.

At the AM-GM point, x^4 = k = 2√2, 2y^2 = 2*(k/2) = k = 2√2, and z^4 = k = 2√2.

So, all terms are equal to k = 2√2.

Therefore, the AM-GM equality condition is satisfied.

But why does it not satisfy the Lagrange equations?

Wait, this suggests that there is a mistake in the calculation of λ from the third equation.

Let me recalculate λ from the third equation.

Given x ≈ 1.296, y ≈ 1.189, z ≈ 1.296

Compute 16z³ / (xy):

First, z³ = (1.296)^3 ≈ 2.177

16 * 2.177 ≈ 34.83

xy = 1.296 * 1.189 ≈ 1.541

So, 34.83 / 1.541 ≈ 22.6

But from the first equation, λ ≈ 5.65.

This inconsistency suggests that this point does not satisfy all three Lagrange equations, which is a contradiction.

But why?

Wait, because when I applied AM-GM, I split the expression into 7 terms, but the Lagrange equations are derived from the original function and constraint.

This suggests that there is a mistake in the AM-GM approach.

Wait, let me check the AM-GM calculation again.

The AM-GM inequality says that the arithmetic mean is greater than or equal to the geometric mean.

But in this case, the arithmetic mean is [x^4 + 4y^2 + 4z^4]/7, and the geometric mean is [x^4 * (2y^2)^2 * (z^4)^4]^{1/7} = [x^4 * 4y^4 * z^{16}]^{1/7}.

But if all terms are equal, then x^4 = 2y^2 = z^4 = k, so x^4 = k, 2y^2 = k ⇒ y^2 = k/2, z^4 = k.

Then, the product is x^4 * 4y^4 * z^{16} = k * 4*(k/2)^2 * k^4 = k * 4*(k²/4) * k^4 = k * k² * k^4 = k^7.

Therefore, the geometric mean is [k^7]^{1/7} = k.

Therefore, AM-GM says that [x^4 + 4y^2 + 4z^4]/7 ≥ k ⇒ x^4 + 4y^2 + 4z^4 ≥ 7k.

But in this case, x^4 + 4y^2 + 4z^4 = k + 2k + k = 4k.

So, 4k ≥ 7k ⇒ 4 ≥ 7, which is false.

Wait, this is a contradiction!

This means that my AM-GM application was incorrect.

Wait, what's the issue here?

Ah! I think I made a mistake in the number of terms.

When I split the expression into 7 terms, I said: x^4, 2y^2, 2y^2, z^4, z^4, z^4, z^4.

But the original expression is x^4 + 4y^2 + 4z^4.

So, splitting 4y^2 into two terms of 2y^2 each, and 4z^4 into four terms of z^4 each.

But the number of terms is 1 + 2 + 4 = 7.

But the AM-GM inequality says that the arithmetic mean of these 7 terms is greater than or equal to the geometric mean.

But if all terms are equal, then the arithmetic mean is equal to the geometric mean.

But in this case, if all terms are equal to k, then the arithmetic mean is k, and the geometric mean is also k.

But the arithmetic mean is [x^4 + 4y^2 + 4z^4]/7 = [k + 2k + k]/7 = 4k/7.

Wait, no! Wait, if all terms are equal to k, then the sum is 7k, so the arithmetic mean is 7k / 7 = k.

But the original expression is x^4 + 4y^2 + 4z^4 = k + 2k + k = 4k.

Wait, this is a contradiction.

Wait, no! Wait, the original expression is x^4 + 4y^2 + 4z^4.

But when I split it into 7 terms, it's x^4 + 2y^2 + 2y^2 + z^4 + z^4 + z^4 + z^4.

If all these terms are equal to k, then the sum is 7k, so the original expression is 7k.

But earlier, I said that x^4 + 4y^2 + 4z^4 = 4k.

This is the mistake!

Ah! Here is the error.

If I split the expression into 7 terms, each equal to k, then the original expression is 7k.

But earlier, I thought that x^4 + 4y^2 + 4z^4 = k + 2k + k = 4k, which is incorrect.

Actually, if x^4 = k, 2y^2 = k, and z^4 = k, then the original expression is x^4 + 4y^2 + 4z^4 = k + 4*(k/2) + 4k = k + 2k + 4k = 7k.

Ah! So, the original expression is 7k, not 4k.

Therefore, the AM-GM inequality says that 7k ≥ 7k, which is equality.

Therefore, the minimum value is 7k, and from the constraint, we can find k.

Wait, let me re-express this.

If all terms are equal to k, then the original expression is 7k.

But from the constraint, we have xyz = 2.

Let me find k in terms of the constraint.

From the AM-GM equality conditions: x^4 = 2y^2 = z^4 = k.

So, x^4 = k, z^4 = k ⇒ x = z.

Also, 2y^2 = k ⇒ y^2 = k/2.

From the constraint: xyz = x * y * z = x^2 * y = 2.

Since x^4 = k, x^2 = sqrt(k).

And y = sqrt(k/2).

So, sqrt(k) * sqrt(k/2) = 2 ⇒ sqrt(k * k / 2) = 2 ⇒ sqrt(k² / 2) = 2 ⇒ k / sqrt(2) = 2 ⇒ k = 2√2.

Therefore, the original expression is 7k = 7 * 2√2 = 14√2 ≈ 19.798.

Wait, but earlier, when I calculated the expression at this point, I got approximately 11.31, but that was a mistake.

Actually, the expression is 7k = 7 * 2√2 ≈ 19.798, which is higher than 16.

So, this resolves the contradiction.

Earlier, I mistakenly thought that the expression was 4k, but it is actually 7k.

Therefore, the AM-GM approach gives a higher value than the Lagrange multiplier method, which suggests that the minimum is indeed 16.

This shows the importance of careful calculation.

Therefore, the correct minimum value is 16, as found by the Lagrange multipliers.

Thus, the minimum value of the expression is 16, achieved at x = √2, y = √2, z = 1.

**Final Answer**
\boxed{16}

To find the minimum value of the expression
$$
x^4 + 4y^2 + 4z^4
$$
under the constraint
$$
xyz = 2,
$$
where $ x, y, z > 0 $, we can use the method of **Lagrange multipliers**.

---

### Step 1: Define the Objective and Constraint

Let
$$
f(x, y, z) = x^4 + 4y^2 + 4z^4
$$
be the function to minimize, and
$$
g(x, y, z) = xyz - 2 = 0
$$
be the constraint.

---

### Step 2: Compute Gradients

We compute the gradients of $ f $ and $ g $:

- $ \nabla f = (4x^3, 8y, 16z^3) $
- $ \nabla g = (yz, xz, xy) $

According to the method of Lagrange multipliers, we set:
$$
\nabla f = \lambda \nabla g
$$
This gives the system of equations:
$$
\begin{cases}
4x^3 = \lambda yz \\
8y = \lambda xz \\
16z^3 = \lambda xy \\
xyz = 2
\end{cases}
$$

---

### Step 3: Solve the System

From the first two equations:
$$
\frac{4x^3}{yz} = \frac{8y}{xz} \Rightarrow 4x^4 z = 8y^2 z \Rightarrow x^4 = 2y^2
$$

From the second and third equations:
$$
\frac{8y}{xz} = \frac{16z^3}{xy} \Rightarrow 8x y^2 = 16x z^4 \Rightarrow y^2 = 2z^4
$$

Now substitute $ y^2 = 2z^4 $ into $ x^4 = 2y^2 $:
$$
x^4 = 2(2z^4) = 4z^4 \Rightarrow x = \sqrt{2} z
$$

Now substitute $ x = \sqrt{2} z $ and $ y^2 = 2z^4 $ into the constraint $ xyz = 2 $:
$$
(\sqrt{2} z)(\sqrt{2} z^2)(z) = 2 \Rightarrow 2z^4 = 2 \Rightarrow z^4 = 1 \Rightarrow z = 1
$$

Then:
- $ x = \sqrt{2} $
- $ y = \sqrt{2} $

---

### Step 4: Evaluate the Expression

Now substitute $ x = \sqrt{2}, y = \sqrt{2}, z = 1 $ into the original expression:
$$
x^4 + 4y^2 + 4z^4 = (\sqrt{2})^4 + 4(\sqrt{2})^2 + 4(1)^4 = 4 + 8 + 4 = 16
$$

---

### Step 5: Verify Minimality

Since the function $ f(x, y, z) $ is smooth and the constraint defines a compact set (in the positive reals), and the function tends to infinity as any variable approaches 0 or infinity, the critical point found is indeed the global minimum.

---

### Final Answer

$$
\boxed{16}
$$

Solution:
By AM-GM,
\begin{align*}
x^4 + 4y^2 + 4z^4 &= x^4 + 2y^2 + 2y^2 + 4z^4 \\
&\ge 4 \sqrt[4]{(x^4)(2y^2)(2y^2)(4z^4)} \\
&= 8xyz \\
&= 16.
\end{align*}Equality occurs when $x^4 = 2y^2 = 4z^2.$  Using the condition $xyz = 2,$ we can solve to get $x = y = \sqrt{2}$ and $z = 1,$ so the minimum value is $\boxed{16}.$