Part (a): Critical values of $f(x)$

The graph provided is of $f'(x)$, which represents the derivative of $f(x)$. The critical values of $f(x)$ occur where $f'(x) = 0$ or where $f'(x)$ is undefined. From the graph, the derivative $f'(x)$ crosses the x-axis (i.e., $f'(x) = 0$) at the following points:

1. $x = -4$
2. $x = 0$
3. $x = 3$

These are the critical points of $f(x)$.

Part (b): Intervals where $f(x)$ is increasing

The function $f(x)$ is increasing when $f'(x) > 0$, meaning the derivative is positive. Based on the graph, $f'(x) > 0$ on the following intervals:

• $(-\infty, -4)$
• $(0, 3)$
• $(3, \infty)$

Part (c): Intervals where $f(x)$ is decreasing

The function $f(x)$ is decreasing when $f'(x) < 0$, meaning the derivative is negative. From the graph, $f'(x) < 0$ on the interval:

• $(-4, 0)$

Part (d): Local maxima of $f(x)$

A local maximum occurs where $f'(x)$ changes from positive to negative. From the graph, this occurs at:

• $x = -4$

Part (e): Local minima of $f(x)$

A local minimum occurs where $f'(x)$ changes from negative to positive. From the graph, this happens at:

• $x = 0$
• $x = 3$

Would you like further details or clarifications on any part? Here are some follow-up questions to consider:

1. How can we use the second derivative to confirm the nature of critical points?
2. How do the concavity and inflection points of $f(x)$ relate to $f'(x)$?
3. Can you determine the behavior of $f(x)$ at the endpoints of the intervals?
4. What happens to $f(x)$ as $x \to \infty$ and $x \to -\infty$?
5. How could we sketch the rough shape of $f(x)$ based on this derivative graph?

Tip: Critical points are often just the start! Always examine the derivative's sign changes to understand the full behavior of the function.