Let’s examine each part of the question one by one based on the provided graph of $f'(x)$, the derivative of $f(x)$.

(a) On what intervals is $f(x)$ increasing? Decreasing?

For $f(x)$ to be increasing, $f'(x)$ must be positive (above the x-axis). Similarly, $f(x)$ is decreasing where $f'(x)$ is negative (below the x-axis).

From the graph:

• $f(x)$ is increasing on the interval $(-2, -1) \cup (2, \infty)$ where $f'(x) > 0$.
• $f(x)$ is decreasing on the interval $(-\infty, -2) \cup (-1, 2)$ where $f'(x) < 0$.

(b) Locate the $x$-value(s) where $f(x)$ has a local maximum and minimum.

Local maxima and minima of $f(x)$ occur where $f'(x) = 0$ (crosses the x-axis) and changes sign:

• At $x = -2$: $f'(x)$ changes from negative to positive, indicating a local minimum of $f(x)$.
• At $x = -1$: $f'(x)$ changes from positive to negative, indicating a local maximum of $f(x)$.
• At $x = 2$: $f'(x)$ changes from negative to positive, indicating another local minimum of $f(x)$.

Thus:

• Local maximum at $x = -1$
• Local minima at $x = -2$ and $x = 2$

(c) On what intervals is $f(x)$ concave upward? Concave downward?

Concavity of $f(x)$ depends on the derivative $f'(x)$:

• $f(x)$ is concave upward where $f'(x)$ is increasing (positive slope).
• $f(x)$ is concave downward where $f'(x)$ is decreasing (negative slope).

From the graph:

• $f(x)$ is concave upward on the intervals $(-\infty, 0) \cup (1, \infty)$ where $f'(x)$ has a positive slope.
• $f(x)$ is concave downward on the interval $(0, 1)$ where $f'(x)$ has a negative slope.

(d) Locate the $x$-value(s) where $f(x)$ has an inflection point.

Inflection points occur where the concavity changes, which is where the slope of $f'(x)$ changes from positive to negative or vice versa.

From the graph:

• The concavity changes at $x = 0$ and $x = 1$.

Thus, the inflection points of $f(x)$ are at $x = 0$ and $x = 1$.

Would you like further clarification on any part? Here are some related questions that expand on these concepts:

1. What does the graph of $f(x)$ look like based on the information given about $f'(x)$?
2. How would changes in $f'(x)$'s behavior affect the second derivative $f''(x)$ and, consequently, the concavity of $f(x)$?
3. Can we determine the relative sizes of the maximum and minimum values of $f(x)$ at the identified points?
4. How would the intervals where $f(x)$ is increasing or decreasing change if $f'(x)$ was shifted vertically?
5. What impact would adding a constant to $f(x)$ have on the locations of maxima, minima, and inflection points?

Tip: Remember that inflection points in a function $f(x)$ occur where the second derivative $f''(x)$ changes sign, corresponding to changes in the slope of $f'(x)$ in this context.