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Using the graph​ shown, describe the shape of the graph of the cost function​ C(x) as x increases from 0 to 4​,000 bottles per month. 
Graph the cost function for 0less than or equalsxless than or equals4. ​[Check the shape of the graph relative to the analysis in part​ (A).] Choose the correct graph below.

To solve this problem, we need to analyze the given graph of \( C'(x) \), the derivative of the cost function \( C(x) \), and use it to describe the shape of the cost function \( C(x) \) as \( x \) increases from 0 to 4,000 bottles per month.

### Analysis

1. **Understanding \( C'(x) \):**
   - The graph provided in the second image is of \( C'(x) \), which shows the rate of change of the cost function \( C(x) \).
   - A positive \( C'(x) \) means \( C(x) \) is increasing.
   - A negative \( C'(x) \) means \( C(x) \) is decreasing.
   - The graph of \( C'(x) \) is a parabola opening upwards with roots at \( x = 0 \) and \( x = 4 \).

2. **Behavior of \( C(x) \):**
   - **From \( x = 0 \) to \( x = 2 \):** \( C'(x) \) is negative, indicating \( C(x) \) is decreasing.
   - **At \( x = 2 \):** \( C'(x) = 0 \), which indicates a local minimum for \( C(x) \).
   - **From \( x = 2 \) to \( x = 4 \):** \( C'(x) \) is positive, indicating \( C(x) \) is increasing.

3. **Shape of \( C(x) \):**
   - \( C(x) \) decreases initially from \( x = 0 \) to \( x = 2 \), reaching a minimum at \( x = 2 \).
   - Then \( C(x) \) increases from \( x = 2 \) to \( x = 4 \).

### Selecting the Correct Graph

Based on the analysis, the correct graph of \( C(x) \) should:
- Start at a higher point when \( x = 0 \),
- Decrease until \( x = 2 \),
- Reach a minimum at \( x = 2 \),
- Then increase until \( x = 4 \).

Given the options:
- **Option A**: This graph seems to match the description as it shows a decreasing cost function from \( x = 0 \) to \( x = 2 \), and then an increasing trend from \( x = 2 \) to \( x = 4 \).

**Conclusion:** The correct graph that matches this behavior is **Option A**.

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Do you have any questions or would you like further details on this analysis?

Here are some related questions to expand on this topic:

1. How does the derivative \( C'(x) \) help in understanding the behavior of a function \( C(x) \)?
2. What does it mean when the derivative \( C'(x) \) changes sign?
3. How can you find local minima and maxima using the derivative graph?
4. Why is the understanding of the cost function important in production analysis?
5. How would the graph of \( C(x) \) look if \( C'(x) \) had a different shape, such as a linear function?
6. Can you explain how the second derivative \( C''(x) \) relates to the concavity of \( C(x) \)?
7. What other real-world situations might involve analyzing cost functions like \( C(x) \)?
8. How might the graph of the cost function change if additional fixed costs were introduced?

**Tip:** When analyzing the behavior of a function using its derivative, pay attention to where the derivative is zero or changes sign, as these points often indicate important changes in the function's behavior, such as local minima, maxima, or inflection points.

Understand how to analyze the shape of a cost function graph using derivatives. This detailed guide covers the behavior of cost functions as x increases from 0 to 4,000 bottles per month. Learn about derivative analysis, local minima and maxima, and practical implications in production analysis.

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