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The image contains a cost function problem from the ThermoMaster Company regarding the production of thermometers. Here's a breakdown of the problem:

Given total cost function:  
\[
C(x) = 6400 + 3x
\]

### (a) Find the average cost function \( \overline{C}(x) \).

The average cost function is defined as the total cost divided by the number of units \(x\). So,

\[
\overline{C}(x) = \frac{C(x)}{x} = \frac{6400 + 3x}{x} = \frac{6400}{x} + 3
\]

### (b) Find the marginal average cost function \( \overline{C}'(x) \).

To find the marginal average cost, we differentiate the average cost function \( \overline{C}(x) \) with respect to \(x\):

\[
\overline{C}'(x) = \frac{d}{dx} \left( \frac{6400}{x} + 3 \right)
\]
\[
\overline{C}'(x) = -\frac{6400}{x^2}
\]

### (c) Interpret the results.

We now interpret the marginal average cost function \( \overline{C}'(x) = -\frac{6400}{x^2} \). This function is always negative for \(x > 0\) because the term \(-\frac{6400}{x^2}\) is negative as \(x^2\) is positive for all \(x > 0\).

So, the correct interpretation is:
- "Since the marginal average cost function is negative for \( x > 0 \), the rate of change of the average cost function is negative for all \( x > 0 \)."

This corresponds to the first option in part (c).

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Do you want more details on this solution or have any specific questions?  
Here are 5 related questions that may help you further:

1. What is the difference between total cost and marginal cost functions?
2. How is the average cost function derived from the total cost function?
3. Why is the marginal average cost function negative in this case?
4. How does the shape of the marginal cost curve affect production decisions?
5. What are real-world applications of interpreting marginal cost functions?

**Tip**: When calculating marginal costs or average costs, remember that the average cost can decrease even if the total cost increases due to economies of scale.

The management of ThermoMaster Company, whose Mexican subsidiary manufactures an indoor-outdoor thermometer, has estimated that the total weekly cost (in dollars) for producing x thermometers is represented by the following function: C(x) = 6400 + 3x. (a) Find the average cost function. (b) Find the marginal average cost function. (c) Interpret your results.

Learn how to calculate the average cost and marginal average cost for the production of thermometers using the cost function C(x) = 6400 + 3x. This problem covers key concepts in cost functions, differentiation, and economic interpretation of cost functions. Suitable for undergraduate economics or calculus students.

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