Let's analyze the graph and address each part of the question.

1. Critical numbers of $f$

Critical numbers occur where the derivative $f'(x)$ is zero or undefined. From the graph, these are points where the curve changes direction (local maxima, minima, or points where the slope is zero).

Based on the curve:

• $f$ changes direction around $x = -1$ and $x = 1$.
• The slope flattens (potential maxima or minima) at these points.

Thus, the critical numbers appear to be $x = -1, 1$.

2. Intervals of increasing and decreasing

To identify intervals of increase or decrease, we examine the slope of the graph:

• From $x = -\infty$ to $x = -1$, the function is decreasing.
• From $x = -1$ to $x = 0$, the function is increasing.
• From $x = 0$ to $x = 1$, the function is decreasing.
• From $x = 1$ onward, the function is increasing.

So:

• Increasing: $(-1, 0) \cup (1, \infty)$
• Decreasing: $(-\infty, -1) \cup (0, 1)$

3. Relative maxima and minima

Relative extrema occur at critical numbers where the function switches from increasing to decreasing (maxima) or vice versa (minima):

• At $x = -1$, the function switches from decreasing to increasing, suggesting a relative minimum.
• At $x = 1$, the function switches from increasing to decreasing, suggesting a relative maximum.

So:

• Relative minimum at $x = -1$.
• Relative maximum at $x = 1$.

Summary:

1. Critical numbers: $x = -1, 1$.
2. Increasing on $(-1, 0) \cup (1, \infty)$; decreasing on $(-\infty, -1) \cup (0, 1)$.
3. Relative minimum at $x = -1$; relative maximum at $x = 1$.

Would you like further details or clarification? Here are five follow-up questions to expand your understanding:

1. How do we determine the sign of the first derivative from a graph?
2. What is the significance of critical points in calculus?
3. How would concavity affect the identification of maxima and minima?
4. How can you test for relative extrema using the second derivative?
5. What is the difference between relative and absolute extrema?

Tip: Always check the behavior of the function near critical points to determine whether they correspond to maxima, minima, or saddle points!