To determine the maximal intervals on which $f$ is increasing or decreasing, we need to analyze the behavior of $f'$ based on its graph.

Observations:

1. $f$ is increasing where $f'(x) > 0$ (above the $x$-axis).
2. $f$ is decreasing where $f'(x) < 0$ (below the $x$-axis).
3. At points where $f'(x) = 0$, the function $f$ has critical points (possible local maxima or minima).

Analysis of the graph:

Intervals of Increase:

• From $x \in (-1, 0)$: $f'(x) > 0$, so $f$ is increasing.
• From $x \in (2, 4)$: $f'(x) > 0$, so $f$ is increasing.

Intervals of Decrease:

• From $x \in (0, 2)$: $f'(x) < 0$, so $f$ is decreasing.

Final Answer:

• Increasing intervals: $(-1, 0) \cup (2, 4)$
• Decreasing intervals: $(0, 2)$

Would you like further clarification or an explanation of critical points?

Here are some related questions you might consider:

1. What happens to $f(x)$ at the critical points where $f'(x) = 0$?
2. How can we confirm whether these intervals are global maxima or minima?
3. Can $f$ have inflection points in this scenario? Why or why not?
4. What is the significance of $f'(x)$ being exactly zero at some points?
5. How does the behavior of $f'(x)$ inform us about the concavity of $f(x)$?

Tip: Always check the sign of $f'(x)$ on