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?

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.