<Question>
Let $a$ and $b$ be positive real numbers such that $ab^2 = 5.$  Find the minimum value of $a^4 + b^8.$
</Question>

<think>
<Plan_1>
Okay, so I need to find the minimum value of \( a^4 + b^8 \) given that \( a \) and \( b \) are positive real numbers and \( ab^2 = 5 \). Hmm, let's see. I remember that optimization problems with constraints can often be solved using methods like substitution or maybe Lagrange multipliers. But since this seems like a problem that might be approachable with AM-GM inequality or maybe calculus, I need to figure out which method would be more straightforward here.
First, let's parse the problem. We have two variables, \( a \) and \( b \), both positive. The constraint is \( ab^2 = 5 \), and we need to minimize \( a^4 + b^8 \). So, the expression to minimize is a sum of two terms, each involving powers of \( a \) and \( b \). The constraint relates \( a \) and \( b \) multiplicatively.

One approach is to express one variable in terms of the other using the constraint and then substitute into the expression we need to minimize. Let's try that.

From the constraint \( ab^2 = 5 \), we can solve for \( a \): \( a = \frac{5}{b^2} \). Then substitute this into the expression \( a^4 + b^8 \). Let's do that.

<prompt_1.1> Substitute \( a = \frac{5}{b^2} \) into the expression \( a^4 + b^8 \) and simplify. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Substituting \( a = \frac{5}{b^2} \) into \( a^4 + b^8 \), we get:

\[
\left( \frac{5}{b^2} \right)^4 + b^8 = \frac{5^4}{b^8} + b^8 = \frac{625}{b^8} + b^8
\]

So now, the problem reduces to minimizing the function \( f(b) = \frac{625}{b^8} + b^8 \) where \( b > 0 \).
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Hmm, okay. Let's denote \( y = b^8 \), but maybe calculus is the way to go here. Let's compute the derivative of \( f(b) \) with respect to \( b \) and set it to zero to find critical points.

First, compute \( f'(b) \):

<prompt_2.1> Compute the derivative of \( f(b) = \frac{625}{b^8} + b^8 \) with respect to \( b \) and set it to zero to find critical points. </prompt_2.1>
-----

Alternatively, maybe I can use the AM-GM inequality on the terms \( \frac{625}{b^8} \) and \( b^8 \). The AM-GM inequality states that for non-negative real numbers, the arithmetic mean is at least the geometric mean. So,

<prompt_2.2> Use the AM-GM inequality on the terms \( \frac{625}{b^8} \) and \( b^8 \) to verify the minimum value. </prompt_2.2>

But wait, just to be thorough, let's check the second derivative to confirm that this critical point is indeed a minimum.

Compute the second derivative \( f''(b) \):

<prompt_2.3> Compute the second derivative \( f''(b) \) and evaluate it at \( b = \sqrt[4]{5} \) to confirm it's a minimum. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
\[
f(b) = 625 b^{-8} + b^8
\]
\[
f'(b) = -8 \times 625 b^{-9} + 8 b^7 = -\frac{5000}{b^9} + 8 b^7
\]

Set the derivative equal to zero:

\[
-\frac{5000}{b^9} + 8 b^7 = 0
\]

Let's solve for \( b \). Move one term to the other side:

\[
8 b^7 = \frac{5000}{b^9}
\]

Multiply both sides by \( b^9 \):

\[
8 b^{16} = 5000
\]

Divide both sides by 8:

\[
b^{16} = \frac{5000}{8} = 625
\]

So,

\[
b^{16} = 625
\]

Take the 16th root of both sides. Since \( b > 0 \), we can write:

\[
b = 625^{1/16}
\]

Now, 625 is 5^4, so:

\[
625^{1/16} = (5^4)^{1/16} = 5^{4/16} = 5^{1/4} = \sqrt[4]{5}
\]

So, \( b = \sqrt[4]{5} \).
</execution_2.1>

<execution_2.2>
\[
\frac{\frac{625}{b^8} + b^8}{2} \geq \sqrt{\frac{625}{b^8} \cdot b^8} = \sqrt{625} = 25
\]
Therefore,
\[
\frac{625}{b^8} + b^8 \geq 50
\]
with equality when \( \frac{625}{b^8} = b^8 \), which implies \( b^{16} = 625 \). Since \( 625 = 5^4 \), so \( b^{16} = 5^4 \), which gives \( b = 5^{4/16} = 5^{1/4} = \sqrt[4]{5} \). Then, \( a = \frac{5}{b^2} = \frac{5}{(5^{1/4})^2} = \frac{5}{5^{1/2}} = 5^{1 - 1/2} = 5^{1/2} = \sqrt{5} \). So, substituting back, \( a = \sqrt{5} \) and \( b = \sqrt[4]{5} \), which gives the minimum value of 50.
</execution_2.2>

<execution_2.3>
First, we need the first derivative again:

\( f'(b) = -\frac{625 \times 8}{b^9} + 8b^7 \)

So,

\( f''(b) = \frac{625 \times 8 \times 9}{b^{10}} - 56b^6 \)

Simplify:

\( f''(b) = \frac{46875}{b^{10}} - 56b^6 \)

Now, substitute \( b = \sqrt[4]{5} \). Let's compute each term:

First term: \( \frac{46875}{(\sqrt[4]{5})^{10}} = \frac{46875}{5^{10/4}} = \frac{46875}{5^{2.5}} = \frac{46875}{5^2 \times 5^{0.5}} = \frac{46875}{25 \times \sqrt{5}} = \frac{46875}{25 \times \sqrt{5}} = \frac{1875}{\sqrt{5}} \)

Second term: \( 56 \times (\sqrt[4]{5})^6 = 56 \times 5^{6/4} = 56 \times 5^{1.5} = 56 \times 5 \times \sqrt{5} = 56 \times 5 \times \sqrt{5} = 280\sqrt{5} \)

So, \( f''(b) = \frac{1875}{\sqrt{5}} - 280\sqrt{5} \). Let's compute this numerically to see if it's positive.

First, \( \frac{1875}{\sqrt{5}} \approx \frac{1875}{2.23607} \approx 838.52 \)

Second term: \( 280 \times 2.23607 \approx 626.10 \)

So, \( f''(b) \approx 838.52 - 626.10 = 212.42 > 0 \). Therefore, the function is concave upwards at this point, confirming that it's a minimum.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Now, we need to compute the minimum value by substituting \( b = \sqrt[4]{5} \) back into the expression \( \frac{625}{b^8} + b^8 \).

Let's compute each term:

<prompt_3.1> Compute the minimum value by substituting \( b = \sqrt[4]{5} \) back into the expression \( \frac{625}{b^8} + b^8 \). </prompt_3.1>
-----

Based on execution_2.2:
But let me verify once again.

Alternatively, could there be a miscalculation? Let's re-examine the steps.

<prompt_3.2> Re-examine the steps to verify the calculations and confirm the minimum value. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
First, \( b^8 = (\sqrt[4]{5})^8 = (5^{1/4})^8 = 5^{8/4} = 5^2 = 25 \). Therefore, \( \frac{625}{b^8} = \frac{625}{25} = 25 \). So, adding the two terms: \( 25 + 25 = 50 \). Therefore, the minimum value of \( a^4 + b^8 \) is 50.
</execution_3.1>

<execution_3.2>
Starting from the substitution: \( a = 5 / b^2 \), so \( a^4 = (5 / b^2)^4 = 625 / b^8 \). Then, \( b^8 \) is just \( b^8 \). So, the expression is \( 625 / b^8 + b^8 \). Then taking derivative: correct. Solving for \( b \): correct. Then AM-GM gives the same result, which is reassuring. So, seems solid.

Alternatively, maybe check with sample values. Suppose we take \( b = 1 \), then \( a = 5 / 1 = 5 \). Then \( a^4 + b^8 = 625 + 1 = 626 \), which is way higher than 50. If we take \( b = \sqrt{5} \), then \( b^2 = 5 \), so \( a = 5 / 5 = 1 \). Then \( a^4 + b^8 = 1 + ( \sqrt{5} )^8 \). Wait, \( (\sqrt{5})^8 = (5^{1/2})^8 = 5^4 = 625 \). So, total is 1 + 625 = 626 again. So, same as before. If we take \( b = \sqrt[4]{5} \), which is approximately 1.495, then \( b^2 = \sqrt{5} \approx 2.236 \), so \( a = 5 / \sqrt{5} = \sqrt{5} \approx 2.236 \). Then \( a^4 = ( \sqrt{5} )^4 = 25 \), and \( b^8 = ( \sqrt[4]{5} )^8 = 5^2 = 25 \). So, 25 + 25 = 50. So, that checks out. If we pick another value, say \( b = 2 \), then \( a = 5 / 4 \), so \( a^4 = (5/4)^4 = 625 / 256 ≈ 2.441 \), and \( b^8 = 256 \), so total ≈ 2.441 + 256 ≈ 258.441, which is bigger than 50. If we take \( b = \sqrt[4]{5} \approx 1.495 \), which gives the minimal point, then the value is 50. So, all checks indicate that 50 is indeed the minimal value.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Wait, but let me check this again step by step to make sure I didn't make a mistake.

<prompt_4.1> Recheck the calculations for the minimum value to ensure no mistakes were made. </prompt_4.1>
-----

Based on execution_3.2:
Therefore, after thorough analysis using substitution and calculus, as well as verification with the AM-GM inequality and testing sample values, the minimum value of \( a^4 + b^8 \) under the constraint \( ab^2 = 5 \) is indeed 50. I feel confident about this answer now.

**Final Answer**
The minimum value of \( a^4 + b^8 \) is \boxed{50}.
</Plan_4>

<Execution_4>
<execution_4.1>
First, confirm \( b = \sqrt[4]{5} \). Then, \( b^2 = (\sqrt[4]{5})^2 = \sqrt{5} \), so \( a = 5 / \sqrt{5} = \sqrt{5} \). Then, \( a^4 = (\sqrt{5})^4 = (5^{1/2})^4 = 5^{2} = 25 \). \( b^8 = (\sqrt[4]{5})^8 = 5^{2} = 25 \). So, 25 + 25 = 50.
</execution_4.1>
</Execution_4>

<Final_answer>
Therefore, the minimum value of $a^4 + b^8$ under the given constraint is $\boxed{50}$.
</Final_answer>
</think>