Let's analyze the given graph of the function $f(x)$ to answer each question.

(a) Critical Values

Critical values occur where the derivative $f'(x)$ is zero or undefined, which typically corresponds to peaks, valleys, or points where the slope changes abruptly. Based on the graph, these appear to be:

• $x = -3$: The function has a point where the slope is zero.
• $x = -1$: There is another change in slope here.
• $x = 0$: The function's slope is zero at this point.
• $x = 3$: Another point where the slope is zero.

So, the critical values are $x = -3, -1, 0, 3$.

(b) Intervals of Increase

A function is increasing on intervals where the graph moves upward as you go from left to right. Observing the graph:

• The function is increasing on $(-\infty, -3)$.
• It increases again on $(-1, 0)$.
• Finally, it increases on $(3, \infty)$.

So, $f(x)$ is increasing on the intervals $(-\infty, -3) \cup (-1, 0) \cup (3, \infty)$.

(c) Intervals of Decrease

The function is decreasing on intervals where the graph moves downward from left to right:

• The function decreases on $(-3, -1)$.
• It decreases again on $(0, 3)$.

Thus, $f(x)$ is decreasing on $(-3, -1) \cup (0, 3)$.

(d) Local Maxima

Local maxima occur at points where the function changes from increasing to decreasing. Based on the graph, there are no points that fit this description, as there are no peaks in the function.

Therefore, the answer for local maxima is $\text{NONE}$.

(e) Local Minima

Local minima occur where the function changes from decreasing to increasing. Observing the graph, the local minima are at:

• $x = -1$
• $x = 3$

Thus, the $x$-values for the local minima are $x = -1, 3$.

Would you like further details on any part of this analysis?

Here are some related questions to explore further:

1. How are critical points determined mathematically for more complex functions?
2. What is the relationship between the first derivative and the increasing/decreasing intervals?
3. How can we find global extrema if the domain of the function is restricted?
4. What role do second derivatives play in identifying the nature of critical points?
5. How can you apply this analysis to real-world situations, like optimization problems?

Tip: Always examine the behavior of a function as $x \to \pm \infty$ to better understand its long-term trends and overall behavior.