Let's analyze the image and solve the problems:

(a) Find all the local maximum and minimum values:

Looking at the graph, we can identify the local maxima and minima by observing where the graph peaks and dips.

• Local Maximums:

  • At $x = -2$, $y = 1$
  • At $x = 3$, $y = 2$

• Local Minimums:

  • At $x = 0$, $y = -1$
  • At $x = 5$, $y = 0$

(b) Find the intervals of increase and decrease:

• Increasing intervals: The graph is increasing (going up) on the following intervals:

  • From $x = -2$ to $x = 0$
  • From $x = 2$ to $x = 3$

  So, the intervals of increase are: $(-2, 0) \cup (2, 3)$

• Decreasing intervals: The graph is decreasing (going down) on the following intervals:

  • From $x = -\infty$ to $x = -2$
  • From $x = 0$ to $x = 2$
  • From $x = 3$ to $x = \infty$

  So, the intervals of decrease are: $(-\infty, -2) \cup (0, 2) \cup (3, \infty)$

Would you like a further breakdown of any part of this problem? Here are five related questions:

1. How do we determine whether a point is a local maximum or minimum from the graph?
2. Can you explain the behavior of the function at critical points where the derivative equals zero?
3. Why is the function increasing on certain intervals but decreasing on others?
4. What is the difference between local maxima/minima and global maxima/minima?
5. How do we confirm these intervals using the first derivative test?

Tip: Always check for critical points where the derivative is zero or undefined to find potential local extrema.