To determine where the function $f$ is increasing, we analyze its derivative graph. Remember:

• If $f'(x) > 0$ (the derivative is positive), $f(x)$ is increasing.
• If $f'(x) < 0$ (the derivative is negative), $f(x)$ is decreasing.

Steps:

1. Look at where the graph of $f'(x)$ is above the x-axis. In those regions, $f'(x) > 0$, meaning $f$ is increasing.

2. Identify the intervals where the graph is positive:

   • From $x = -4$ to $x = -2$
   • From $x = 1$ to $x = 4$

Final Answer:

The function $f$ is increasing on the intervals: $(-4, -2) \, \cup \, (1, 4)$

Do you want a deeper explanation or additional help with related concepts?

Here are 5 related questions to expand your understanding:

1. What does it mean when $f'(x) = 0$, and how does it relate to critical points?
2. How can you determine where $f$ is decreasing from the derivative graph?
3. What would the original function $f$ look like based on this derivative graph?
4. How can you use the second derivative to determine concavity of $f$?
5. If $f'(x)$ were a straight line, what would that tell us about $f$?

Tip: Always remember that the behavior of $f$ depends on the sign of $f'(x)$: positive for increasing, negative for decreasing!