<Question>
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 2.$  Find the minimum value of
\[x^4 + 4y^2 + 4z^4.\]
</Question>

<think>
<Plan_1>
Okay, so I need to find the minimum value of the expression \( x^4 + 4y^2 + 4z^4 \) given that \( x \), \( y \), and \( z \) are positive real numbers and their product \( xyz = 2 \). Hmm, let's see. I remember that optimization problems with constraints often use methods like Lagrange multipliers, but since all variables are positive, maybe inequalities like AM-GM could also work here. Let me try to think through both approaches.

First, let's recall the AM-GM inequality, which states that for non-negative real numbers, the arithmetic mean is at least the geometric mean. The standard form is \( \frac{a_1 + a_2 + \dots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \dots a_n} \), with equality when all numbers are equal. Maybe I can apply this inequality to the terms in the expression \( x^4 + 4y^2 + 4z^4 \). But the terms have different exponents and coefficients, so it might not be straightforward.

Alternatively, maybe I can manipulate the expression to have terms that can be compared using AM-GM. Let's look at each term:

1. \( x^4 \): This is a fourth power.
2. \( 4y^2 \): This is a squared term with a coefficient.
3. \( 4z^4 \): Another fourth power term with a coefficient.

The constraint is \( xyz = 2 \). Since the variables are positive, maybe I can express one variable in terms of the others using the constraint. For example, \( z = \frac{2}{xy} \). Then substitute this into the original expression to reduce the number of variables.

Let me try that substitution.

<prompt_1.1> Substitute \( z = \frac{2}{xy} \) into the expression \( x^4 + 4y^2 + 4z^4 \) and simplify. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Substituting \( z = \frac{2}{xy} \) into the expression gives:

\[
x^4 + 4y^2 + 4\left( \frac{2}{xy} \right)^4 = x^4 + 4y^2 + 4 \cdot \frac{16}{x^4 y^4} = x^4 + 4y^2 + \frac{64}{x^4 y^4}
\]

Hmm, so now the expression is in terms of \( x \) and \( y \). This might still be tricky. Maybe I can set \( t = x^2 \) and \( s = y^2 \) to simplify the exponents. Let's try that substitution.

Let \( t = x^2 \) and \( s = y^2 \). Then \( x = \sqrt{t} \) and \( y = \sqrt{s} \). The expression becomes:

\[
t^2 + 4s + \frac{64}{t^2 s^2}
\]

And the constraint \( xyz = 2 \) becomes:

\[
\sqrt{t} \cdot \sqrt{s} \cdot z = 2
\]

But \( z = \frac{2}{\sqrt{t} \sqrt{s}} \), so substituting back into the expression might not help. Wait, maybe this substitution complicates things more. Let me think of another approach.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Alternatively, maybe I can apply the AM-GM inequality directly to the three terms \( x^4 \), \( 4y^2 \), and \( 4z^4 \). But to apply AM-GM, the terms should be positive, which they are, and we can try to write them as sums of terms that can be compared. However, the coefficients complicate things. Let me see.

Alternatively, maybe split the terms into multiple parts so that the coefficients can be incorporated into the AM-GM. For example, the term \( 4y^2 \) can be written as \( y^2 + y^2 + y^2 + y^2 \), and similarly, \( 4z^4 \) can be written as \( z^4 + z^4 + z^4 + z^4 \). Then, the entire expression becomes \( x^4 + y^2 + y^2 + y^2 + y^2 + z^4 + z^4 + z^4 + z^4 \). That's 1 term of \( x^4 \), 4 terms of \( y^2 \), and 4 terms of \( z^4 \), totaling 9 terms. Then, applying AM-GM to these 9 terms.

<prompt_2.1> Apply the AM-GM inequality to the split terms and find the condition for equality. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
Wait, but the original expression is \( x^4 + 4y^2 + 4z^4 \), so splitting \( 4y^2 \) into four \( y^2 \) terms and \( 4z^4 \) into four \( z^4 \) terms gives us 1 + 4 + 4 = 9 terms. Then, applying AM-GM to these 9 terms:

\[
\frac{x^4 + y^2 + y^2 + y^2 + y^2 + z^4 + z^4 + z^4 + z^4}{9} \geq \sqrt[9]{x^4 \cdot (y^2)^4 \cdot (z^4)^4}
\]

Simplifying the right-hand side:

\[
\sqrt[9]{x^4 y^8 z^8} = (x^4 y^8 z^8)^{1/9} = x^{4/9} y^{8/9} z^{8/9}
\]

But the left-hand side is \( \frac{x^4 + 4y^2 + 4z^4}{9} \). So,

\[
\frac{x^4 + 4y^2 + 4z^4}{9} \geq x^{4/9} y^{8/9} z^{8/9}
\]

Hmm, but this seems complicated because the exponents on the right-hand side depend on \( x \), \( y \), and \( z \). Maybe this approach isn't the most effective. Let me think of another way.

Alternatively, maybe use weighted AM-GM. Since the exponents in the expression are 4, 2, and 4 for \( x \), \( y \), and \( z \) respectively, and the constraint is \( xyz = 2 \), perhaps we can set up the inequality with weights corresponding to the exponents. Let me try that.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Let me denote the exponents as follows:

- For \( x^4 \), the exponent is 4.
- For \( y^2 \), the exponent is 2.
- For \( z^4 \), the exponent is 4.

So, total exponents sum to 4 + 2 + 4 = 10. Wait, but in AM-GM, the weights should correspond to the coefficients in the product. Alternatively, maybe split the terms such that each term in the product has the same exponent.

<prompt_3.1> Set up the weighted AM-GM inequality for the expression \( x^4 + 4y^2 + 4z^4 \). </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
Alternatively, maybe use the method of Lagrange multipliers here. Let's set up the Lagrangian. Let me recall that for minimizing \( f(x, y, z) = x^4 + 4y^2 + 4z^4 \) subject to the constraint \( g(x, y, z) = xyz - 2 = 0 \), the Lagrangian is:

\[
\mathcal{L}(x, y, z, \lambda) = x^4 + 4y^2 + 4z^4 + \lambda (xyz - 2)
\]

Then, taking partial derivatives and setting them to zero:

1. \( \frac{\partial \mathcal{L}}{\partial x} = 4x^3 + \lambda yz = 0 \)
2. \( \frac{\partial \mathcal{L}}{\partial y} = 8y + \lambda xz = 0 \)
3. \( \frac{\partial \mathcal{L}}{\partial z} = 16z^3 + \lambda xy = 0 \)
4. \( \frac{\partial \mathcal{L}}{\partial \lambda} = xyz - 2 = 0 \)

So, we have four equations:

1. \( 4x^3 + \lambda yz = 0 \)  (1)
2. \( 8y + \lambda xz = 0 \)    (2)
3. \( 16z^3 + \lambda xy = 0 \) (3)
4. \( xyz = 2 \)              (4)
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
From equations (1), (2), and (3), we can express \( \lambda \) in terms of \( x \), \( y \), \( z \).

<prompt_4.1> Solve the system of equations from the Lagrangian to find relationships between \( x \), \( y \), and \( z \). </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
Let's solve for \( \lambda \):

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

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

From (3): \( \lambda = -\frac{16z^3}{xy} \)

Therefore, we can set these expressions equal to each other:

First, set (1) equal to (2):

\[
-\frac{4x^3}{yz} = -\frac{8y}{xz}
\]

Simplify:

Multiply both sides by \( -1 \):

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

Multiply both sides by \( yz \):

\[
4x^3 = \frac{8y}{xz} \cdot yz = \frac{8y^2 z}{x}
\]

Multiply both sides by \( x \):

\[
4x^4 = 8y^2 z
\]

Divide both sides by 4:

\[
x^4 = 2y^2 z \quad \text{(5)}
\]

Now, set (2) equal to (3):

\[
-\frac{8y}{xz} = -\frac{16z^3}{xy}
\]

Simplify:

Multiply both sides by \( -1 \):

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

Multiply both sides by \( xz \):

\[
8y = \frac{16z^3}{xy} \cdot xz = \frac{16z^4}{y}
\]

Multiply both sides by \( y \):

\[
8y^2 = 16z^4
\]

Divide both sides by 8:

\[
y^2 = 2z^4 \quad \text{(6)}
\]

So, from equation (6): \( y^2 = 2z^4 \), which implies \( y = \sqrt{2} z^2 \). Since \( y \) is positive.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
Now, let's substitute \( y = \sqrt{2} z^2 \) into equation (5):

Equation (5): \( x^4 = 2y^2 z \)

Substitute \( y^2 = 2z^4 \):

<prompt_5.1> Substitute \( y = \sqrt{2} z^2 \) into equation (5) and solve for \( x \) in terms of \( z \). </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
\( x^4 = 2 \cdot 2z^4 \cdot z = 4z^5 \)

Therefore, \( x^4 = 4z^5 \), so \( x = (4z^5)^{1/4} = (4)^{1/4} z^{5/4} = \sqrt{2} z^{5/4} \).

So, \( x = \sqrt{2} z^{5/4} \).
</execution_5.1>
</Execution_5>

<Plan_6>
Based on execution_5.1:
Now, we have expressions for \( x \) and \( y \) in terms of \( z \). Let's recall the constraint \( xyz = 2 \). Substitute \( x \), \( y \), and \( z \):

\( x = \sqrt{2} z^{5/4} \)

\( y = \sqrt{2} z^2 \)

\( z = z \)

So, multiplying them together:

<prompt_6.1> Solve for \( z \) using the constraint \( xyz = 2 \) and the expressions for \( x \) and \( y \) in terms of \( z \). </prompt_6.1>
</Plan_6>

<Execution_6>
<execution_6.1>
\( \sqrt{2} z^{5/4} \cdot \sqrt{2} z^2 \cdot z = 2 \)

Multiply the constants:

\( \sqrt{2} \cdot \sqrt{2} = 2 \)

Multiply the exponents of \( z \):

\( z^{5/4} \cdot z^2 \cdot z = z^{5/4 + 2 + 1} = z^{5/4 + 3} = z^{17/4} \)

So, the equation becomes:

\( 2 \cdot z^{17/4} = 2 \)

Divide both sides by 2:

\( z^{17/4} = 1 \)

Taking both sides to the power of 4/17:

\( z = 1^{4/17} = 1 \)

So, \( z = 1 \).
</execution_6.1>
</Execution_6>

<Plan_7>
Based on execution_6.1:
Now, substitute back to find \( y \) and \( x \):

From \( z = 1 \):

<prompt_7.1> Find the values of \( x \), \( y \), and \( z \) and verify if they satisfy the original equations. </prompt_7.1>
</Plan_7>

<Execution_7>
<execution_7.1>
\( y = \sqrt{2} z^2 = \sqrt{2} \cdot 1^2 = \sqrt{2} \)

\( x = \sqrt{2} z^{5/4} = \sqrt{2} \cdot 1^{5/4} = \sqrt{2} \)

So, \( x = \sqrt{2} \), \( y = \sqrt{2} \), \( z = 1 \).

Now, let's check if these values satisfy the original equations.

Check equation (1): \( 4x^3 + \lambda yz = 0 \)

First, compute \( x^3 = (\sqrt{2})^3 = 2^{3/2} = 2 \sqrt{2} \)

So, \( 4x^3 = 4 \cdot 2 \sqrt{2} = 8 \sqrt{2} \)

Compute \( yz = \sqrt{2} \cdot 1 = \sqrt{2} \), so \( \lambda yz = \lambda \sqrt{2} \)

Thus, equation (1): \( 8 \sqrt{2} + \lambda \sqrt{2} = 0 \) => \( \lambda = -8 \)

Similarly, check equation (2): \( 8y + \lambda xz = 0 \)

\( y = \sqrt{2} \), so \( 8y = 8 \sqrt{2} \)

\( xz = \sqrt{2} \cdot 1 = \sqrt{2} \), so \( \lambda xz = -8 \cdot \sqrt{2} \)

Thus, \( 8 \sqrt{2} - 8 \sqrt{2} = 0 \), which holds.

Equation (3): \( 16z^3 + \lambda xy = 0 \)

\( z = 1 \), so \( 16z^3 = 16 \)

\( xy = \sqrt{2} \cdot \sqrt{2} = 2 \), so \( \lambda xy = -8 \cdot 2 = -16 \)

Thus, \( 16 - 16 = 0 \), which also holds.

And the constraint \( xyz = \sqrt{2} \cdot \sqrt{2} \cdot 1 = 2 \), which is satisfied.
</execution_7.1>
</Execution_7>

<Plan_8>
Based on execution_7.1:
So, the critical point is at \( x = \sqrt{2} \), \( y = \sqrt{2} \), \( z = 1 \). Now, let's compute the value of the expression at this point:

<prompt_8.1> Calculate the minimum value of the expression at the critical point. </prompt_8.1>
</Plan_8>

<Execution_8>
<execution_8.1>
\( x^4 + 4y^2 + 4z^4 = (\sqrt{2})^4 + 4 (\sqrt{2})^2 + 4 (1)^4 \)

Calculate each term:

- \( (\sqrt{2})^4 = (2)^{2} = 4 \)
- \( 4 (\sqrt{2})^2 = 4 \cdot 2 = 8 \)
- \( 4 (1)^4 = 4 \cdot 1 = 4 \)

Adding them up: \( 4 + 8 + 4 = 16 \)

So, the value is 16.
</execution_8.1>
</Execution_8>

<Plan_9>
Based on execution_8.1:
Is this the minimum? Well, since the problem is convex? Wait, but we have multiple variables and a constraint, but given that we found a critical point and the function tends to infinity as any variable approaches 0 or infinity (since the product is fixed), this critical point is likely to be the global minimum.
Let me verify the behavior at the boundaries.

<prompt_9.1> Verify the behavior of the expression at the boundaries to confirm the critical point as the global minimum. </prompt_9.1>
</Plan_9>

<Execution_9>
<execution_9.1>
Suppose \( x \) approaches 0, then \( yz \) approaches infinity to maintain \( xyz = 2 \). Then, \( x^4 \) approaches 0, but \( 4y^2 \) and \( 4z^4 \) would approach infinity. Similarly, if \( x \) approaches infinity, then \( y \) and \( z \) approach 0, making \( x^4 \) approach infinity and \( 4y^2 \) and \( 4z^4 \) approach 0. So, the expression tends to infinity in both extremes. Therefore, the critical point we found is indeed the global minimum.
</execution_9.1>
</Execution_9>

<Plan_10>
Based on execution_9.1:
Therefore, the minimum value is 16, achieved when \( x = \sqrt{2} \), \( y = \sqrt{2} \), \( z = 1 \).

Wait, but let me double-check. Let me try another set of values to see if the expression can be lower.

<prompt_10.1> Double-check the minimum value by testing other sets of values for \( x \), \( y \), and \( z \). </prompt_10.1>
</Plan_10>

<Execution_10>
<execution_10.1>
Suppose we take \( x = 1 \), then \( yz = 2 \). Let's set \( y = z = \sqrt{2} \). Then, compute the expression:

\( x^4 = 1 \)

\( 4y^2 = 4 (\sqrt{2})^2 = 8 \)

\( 4z^4 = 4 (\sqrt{2})^4 = 4 \cdot 4 = 16 \)

Total: \( 1 + 8 + 16 = 25 \), which is higher than 16.

Another test: Let me take \( x = 2 \), then \( yz = 1 \). Let me set \( y = z = 1 \). Then:

\( x^4 = 16 \)

\( 4y^2 = 4 \)

\( 4z^4 = 4 \)

Total: \( 16 + 4 + 4 = 24 \), still higher.

Another test: Suppose \( x = \sqrt{2} \), \( y = \sqrt{2} \), \( z = 1 \), which gives 16. Let me slightly perturb \( z \) to see. Let me take \( z = 1.1 \), then \( xyz = 2 \) implies \( x \cdot y = 2 / 1.1 ≈ 1.818 \). Let me set \( x = y = \sqrt{1.818} ≈ 1.348 \). Then compute the expression:

\( x^4 ≈ (1.348)^4 ≈ (1.818)^2 ≈ 3.305 \)

\( 4y^2 ≈ 4*(1.818) ≈ 7.272 \)

\( 4z^4 ≈ 4*(1.1)^4 ≈ 4*(1.4641) ≈ 5.856 \)

Total ≈ 3.305 + 7.272 + 5.856 ≈ 16.433, which is higher than 16.

Alternatively, take \( z = 0.9 \), then \( xyz = 2 \) implies \( x \cdot y = 2 / 0.9 ≈ 2.222 \). Let \( x = y = \sqrt{2.222} ≈ 1.491 \). Then:

\( x^4 ≈ (1.491)^4 ≈ (2.222)^2 ≈ 4.938 \)

\( 4y^2 ≈ 4*(2.222) ≈ 8.888 \)

\( 4z^4 ≈ 4*(0.9)^4 ≈ 4*(0.6561) ≈ 2.624 \)

Total ≈ 4.938 + 8.888 + 2.624 ≈ 16.45, again higher. So, it seems like 16 is indeed the minimum.
</execution_10.1>
</Execution_10>

<Plan_11>
Based on execution_10.1:
Alternatively, let's check if our initial AM-GM approach could be adjusted. Remember, when we tried splitting the terms, we had 9 terms, but the exponents didn't align. However, if we consider the exponents in the original problem, maybe we can apply weighted AM-GM with different weights.

<prompt_11.1> Re-evaluate the expression using weighted AM-GM with different weights. </prompt_11.1>
</Plan_11>

<Execution_11>
<execution_11.1>
Let me recall that the weighted AM-GM inequality states that for positive real numbers \( a_i \) and positive weights \( w_i \) with \( \sum w_i = 1 \), we have:

\[
\prod a_i^{w_i} \leq \sum w_i a_i
\]

But maybe in this case, since the exponents are different, we can split the terms into parts with exponents. Let me think. Alternatively, use the method of Lagrange multipliers as above, which gave a concrete answer.

Alternatively, let me try another approach. Suppose we fix the product \( xyz = 2 \), and we need to minimize \( x^4 + 4y^2 + 4z^4 \). Maybe use substitution variables. Let me set \( a = x^2 \), \( b = 2y \), and \( c = 2z^2 \). Then, the expression becomes \( a^2 + b^2 + c^2 \). Let's see if this helps.

But we need to relate \( a \), \( b \), and \( c \) to the constraint \( xyz = 2 \). Let's express \( x \), \( y \), \( z \) in terms of \( a \), \( b \), \( c \):

- \( x = \sqrt{a} \)
- \( y = \frac{b}{2} \)
- \( z = \sqrt{\frac{c}{2}} \)

Then, the constraint \( xyz = 2 \) becomes:

\[
\sqrt{a} \cdot \frac{b}{2} \cdot \sqrt{\frac{c}{2}} = 2
\]

Simplify:

Multiply the constants:

\[
\frac{1}{2} \cdot \frac{1}{\sqrt{2}} = \frac{1}{2 \sqrt{2}}
\]

Multiply the variables:

\[
\sqrt{a} \cdot b \cdot \sqrt{c} = \sqrt{a c} \cdot b
\]

So, the constraint is:

\[
\frac{b \sqrt{a c}}{2 \sqrt{2}} = 2 \quad \Rightarrow \quad b \sqrt{a c} = 4 \sqrt{2}
\]

Now, the problem is to minimize \( a^2 + b^2 + c^2 \) subject to \( b \sqrt{a c} = 4 \sqrt{2} \). Hmm, maybe this substitution complicates things more. Let's see.

Alternatively, maybe use the Cauchy-Schwarz inequality. The expression \( x^4 + 4y^2 + 4z^4 \) can be thought of as a sum of squares. But I don't see an immediate way to apply Cauchy-Schwarz here.

Alternatively, maybe use the method of substituting variables to reduce the number of variables. For example, since \( xyz = 2 \), we can set \( x = \frac{2}{yz} \), and substitute into the expression. Let's try that.
</execution_11.1>
</Execution_11>

<Final_answer>
Therefore, the minimum value of the expression \(x^4 + 4y^2 + 4z^4\) under the given constraint is \(\boxed{16}\).
</Final_answer>
</think>