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The problem asks to find the minimum value of the function \( f(x, y, z) = x^3 + 12 \frac{yz}{x} + 16 \left( \frac{1}{yz} \right)^{3/2} \), where \( x, y, \) and \( z \) are positive real numbers. Let's find the minimum value of this function.

### Step-by-Step Solution:

1. **Simplify the Function:**
   Rewrite the function in a more manageable form:
   \[
   f(x, y, z) = x^3 + \frac{12yz}{x} + 16 \left( \frac{1}{yz} \right)^{3/2}
   \]

2. **Find Partial Derivatives:**
   To find the critical points, we need to take the partial derivatives of \( f \) with respect to \( x \), \( y \), and \( z \) and set them equal to zero.

   - **Partial Derivative with respect to \( x \):**
     \[
     \frac{\partial f}{\partial x} = 3x^2 - \frac{12yz}{x^2}
     \]
     Set \( \frac{\partial f}{\partial x} = 0 \):
     \[
     3x^4 = 12yz \implies x^4 = 4yz \implies x = \sqrt[4]{4yz}
     \]

   - **Partial Derivative with respect to \( y \):**
     \[
     \frac{\partial f}{\partial y} = \frac{12z}{x} - 24 \left( \frac{1}{yz} \right)^{5/2}
     \]
     Set \( \frac{\partial f}{\partial y} = 0 \).

   - **Partial Derivative with respect to \( z \):**
     \[
     \frac{\partial f}{\partial z} = \frac{12y}{x} - 24 \left( \frac{1}{yz} \right)^{5/2}
     \]
     Set \( \frac{\partial f}{\partial z} = 0 \).

3. **Solve for Critical Points:**
   Solving these equations simultaneously will give us the values of \( x \), \( y \), and \( z \).

4. **Evaluate \( f(x, y, z) \) at the Critical Points:**
   Substitute the critical points back into the function \( f(x, y, z) \) to find the minimum value.

Given the complexity of solving the system analytically, let’s use some numerical methods or appropriate substitutions to find the values that yield the minimum. For the options provided, the correct minimum value is likely found by substituting known values that simplify the expression.

After testing, we find that the minimum value of \( f(x, y, z) \) is **24**, which matches option **(c)**.

Would you like further details on any specific part of the process or have any other questions? Here are some related questions to explore:

1. How do you find critical points of a function with multiple variables?
2. What are the properties of functions that have a minimum or maximum?
3. How do you apply Lagrange multipliers for constrained optimization?
4. Can you explain the concept of gradient descent in multivariable calculus?
5. What numerical methods can be used to find minima of complex functions?

**Tip:** When dealing with optimization problems in multiple variables, always ensure that the critical points you find are within the given domain and check for possible boundary solutions.

Cari nilai minimum dari f(x, y, z) = x^3 + 12 (yz/x) + 16 (1/yz)^(3/2) dimana x, y, dan z adalah bilangan real positif.

This problem focuses on finding the minimum value of the multivariable function f(x, y, z) = x^3 + 12(yz/x) + 16(1/yz)^(3/2) by using partial derivatives and optimization techniques. Suitable for university-level calculus students, this example demonstrates how to find critical points in functions with multiple variables.

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