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The image provides a problem related to cost functions in economics. Here's the relevant information extracted from the image:

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**Problem Statement**:

The economic cost (CT) of a production factor is defined as the payment required to maintain it in its current use. It is also described as the remuneration that this factor would receive in its best alternative employment.

The cost function provided is:

\[
CT = n \cdot x^{-\sqrt{x}}
\]

Where \( n = 9 \), and \( x \) represents the quantity.

The task is to determine the marginal cost function (denoted \( C_mg \)), which is the derivative of the total cost function with respect to \( x \). Additionally, we need to compute the marginal cost when \( x = 7 \).

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### Solution Steps:

1. **Differentiate the cost function** to find the marginal cost function.
2. **Substitute \( x = 7 \)** into the marginal cost function to calculate the marginal cost at this specific quantity.

Let me solve this step-by-step.import sympy as sp

# Define the variables and the cost function
x = sp.symbols('x')
n = 9
CT = n * x**(-sp.sqrt(x))

# Find the marginal cost function (derivative of CT with respect to x)
Cmarginal = sp.diff(CT, x)

# Calculate the marginal cost for x = 7
Cmarginal_at_7 = Cmarginal.subs(x, 7)

Cmarginal, Cmarginal_at_7.simplify() The marginal cost function \( C_{mg}(x) \), derived from the total cost function, is:

\[
C_{mg}(x) = 9 \cdot x^{-\sqrt{x}} \left( - \frac{\ln(x)}{2\sqrt{x}} - \frac{1}{\sqrt{x}} \right)
\]

Substituting \( x = 7 \) into the marginal cost function, the marginal cost at \( x = 7 \) is approximately:

\[
C_{mg}(7) \approx - \frac{9 \ln(7) + 18}{2 \cdot 7^{\sqrt{7} + \frac{1}{2}}}
\]

This is a complex expression involving logarithms and powers of 7. If you'd like, I can calculate a more precise numerical value for this marginal cost.

Would you like to see the numerical result, or do you have any questions? 

Here are five related questions to extend the problem:
1. What is the economic interpretation of the marginal cost function?
2. How does the marginal cost change as \( x \) increases?
3. What happens to the total cost when the marginal cost is negative?
4. Could you explain how logarithmic terms affect the cost calculations?
5. How would the marginal cost function change if \( n \) were different?

**Tip**: When solving cost functions, understanding the behavior of derivatives can help interpret economic decisions more effectively.

Explore the economic cost functions and learn how to calculate marginal costs using advanced mathematical techniques. Includes detailed steps on deriving and interpreting cost and marginal cost functions.

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