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​(A) 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. 
Where is the cost function​ C(x) increasing and​ decreasing? 


The graph provided shows the derivative of the cost function \( C'(x) \), which represents the rate of change of the cost function \( C(x) \) with respect to the number of bottles produced \( x \) (in thousands). 

### Shape of the Graph of \( C(x) \):

1. **Decreasing Interval:**
   - From \( x = 0 \) to \( x = 2,000 \) bottles: 
     - The graph of \( C'(x) \) is positive and decreasing. Since \( C'(x) > 0 \), the cost function \( C(x) \) is increasing. However, because \( C'(x) \) is decreasing, the rate at which the cost is increasing is slowing down.

2. **Minimum Point:**
   - At \( x = 2,000 \) bottles:
     - The graph of \( C'(x) \) reaches its minimum point (which could correspond to the vertex of a parabola if \( C(x) \) is quadratic). Here, \( C'(x) = 0 \), which means that \( C(x) \) has a local minimum at this point. 

3. **Increasing Interval:**
   - From \( x = 2,000 \) to \( x = 4,000 \) bottles:
     - The graph of \( C'(x) \) starts to increase again. Since \( C'(x) \) is negative and increasing towards zero, the cost function \( C(x) \) decreases initially and then increases after reaching the minimum at \( x = 2,000 \).

### Summary:

- **Increasing**: The cost function \( C(x) \) is increasing as \( x \) moves from 0 to 2,000 bottles.
- **Decreasing**: The cost function \( C(x) \) decreases as \( x \) moves from 2,000 to 4,000 bottles.

The cost function \( C(x) \) has a minimum at \( x = 2,000 \) bottles, meaning that \( C(x) \) initially decreases as production increases up to 2,000 bottles, then increases as production continues to 4,000 bottles.

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Would you like more details or have any questions?

Here are 8 questions to consider:

1. What does it mean for a cost function to have a minimum at a certain production level?
2. How would the graph of \( C(x) \) look if \( C'(x) \) were always positive?
3. What could cause the cost function to increase after a certain number of bottles?
4. How would the graph of \( C(x) \) change if the minimum occurred at a different value of \( x \)?
5. What is the significance of the slope of the graph of \( C'(x) \)?
6. Can the cost function \( C(x) \) ever decrease if \( C'(x) > 0 \)?
7. How does the concept of concavity relate to the graph of \( C(x) \)?
8. What could be a real-life scenario where this type of cost function is applicable?

**Tip:** To fully understand the relationship between \( C(x) \) and \( C'(x) \), it's essential to consider how the derivative influences the behavior of the original function.

Explore the behavior of the cost function C(x) graph as production levels vary from 0 to 4,000 bottles per month. Learn about the significance of derivatives in determining where C(x) increases and decreases.

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