Describe the concavity of​ C(x). Choose the correct answer below.

To describe the concavity of \( C(x) \), we need to analyze the derivative of the cost function \( C'(x) \). The concavity of \( C(x) \) is determined by the sign of the second derivative \( C''(x) \).

Given the graph of \( C'(x) \):

- **Concave Up:** If \( C''(x) > 0 \), the graph of \( C(x) \) is concave up (shaped like a "U").
- **Concave Down:** If \( C''(x) < 0 \), the graph of \( C(x) \) is concave down (shaped like an upside-down "U").

From the graph of \( C'(x) \):

1. **From \( x = 0 \) to \( x = 2,000 \) bottles:** 
   - \( C'(x) \) is decreasing, meaning \( C''(x) < 0 \). Therefore, \( C(x) \) is **concave down** in this interval.

2. **From \( x = 2,000 \) to \( x = 4,000 \) bottles:** 
   - \( C'(x) \) is increasing, meaning \( C''(x) > 0 \). Therefore, \( C(x) \) is **concave up** in this interval.

### Conclusion:

The correct description of the concavity of \( C(x) \) is:
- **Concave down** from \( x = 0 \) to \( x = 2,000 \) bottles.
- **Concave up** from \( x = 2,000 \) to \( x = 4,000 \) bottles.

Would you like more details or have any questions?

Here are 8 related questions:

1. What effect does the concavity of \( C(x) \) have on the cost of production?
2. How does a point of inflection relate to the concavity of \( C(x) \)?
3. Can you determine the concavity of \( C(x) \) just by looking at the graph of \( C(x) \)?
4. How would \( C(x) \) behave if it were always concave up?
5. How is the second derivative \( C''(x) \) related to the curvature of \( C(x) \)?
6. What might cause the transition from concave down to concave up in a real-world cost function?
7. Could \( C(x) \) have multiple intervals of concave up or down?
8. How do changes in concavity affect the optimization of production costs?

**Tip:** Always remember that the second derivative tells us about the concavity of a function. Positive second derivatives indicate concave-up behavior, while negative ones indicate concave-down behavior.

Understand how to describe the concavity of C(x) using calculus. Analyze the second derivative C''(x) to determine whether C(x) is concave up or concave down in different intervals. Suitable for college-level students and anyone interested in calculus and optimization.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation