https://file.aiquickdraw.com/mathbot/file/1730747886281jupq4o3m.jpg

Let's analyze the problem using the information from the graph.

### Part (A)
You need to find intervals where \( f'(x) < 0 \), which means the function \( f(x) \) is decreasing.

From the graph, it appears that:
- \( f(x) \) is decreasing in the intervals \( (-\infty, b) \), \( (0, e) \), and \( (g, h) \).
Thus, the correct choice for (A) is:
- **\( (-\infty, b), (0, e), (g, h) \)**.

### Part (B)
Now, find intervals where \( f'(x) > 0 \), meaning \( f(x) \) is increasing.

From the graph, we observe:
- \( f(x) \) is increasing in the intervals \( (b, d) \) and \( (e, \infty) \).
Thus, the correct choice for (B) is:
- **\( (b, d), (e, \infty) \)**.

Would you like further details on the reasoning process?

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Here are some related questions to explore the concepts further:

1. What does the sign of \( f'(x) \) tell us about the behavior of the function \( f(x) \)?
2. How can we determine intervals where \( f''(x) \) is positive or negative based on concavity?
3. How do we find the points where \( f(x) \) has local maxima or minima?
4. What does the second derivative test tell us about concavity and inflection points?
5. How do we interpret the intervals in which \( f'(x) = 0 \)?

**Tip:** For visual problems, sketching the derivative graph can clarify intervals where \( f'(x) \) changes sign.

Use the graph of f in the figure to complete parts (A) and (B) by finding the intervals on which f'(x) < 0 and f'(x) > 0.

Explore how to find the intervals on which f'(x) < 0 and f'(x) > 0 using the graph of f(x). This calculus problem involves analyzing the behavior of a function based on the sign of its first derivative.

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