Let's analyze the graph and the questions from the image to provide the correct answers.

Part (a): Local Maximum and Minimum

You are asked to find the local maxima and minima. Let's break it down:

Local Maximum

1. The first local maximum appears to occur at x ≈ -1. The graph peaks here, and the corresponding y-value is 1. So, the point is: $(-1, 1)$

2. The second local maximum seems to occur at x ≈ 2. Here the graph reaches another peak, and the y-value is 1. The point is: $(2, 1)$

Local Minimum

1. There is one local minimum at x ≈ 0. The graph dips down to y ≈ -1. So, the local minimum is: $(0, -1)$

Part (b): Increasing and Decreasing Intervals

Now, let's identify where the function is increasing and decreasing.

Increasing Intervals

• The graph is increasing between:
  • From x = -∞ to x = -1 (up to the first local maximum at -1).
  • From x = 0 to x = 2 (up to the second local maximum at 2).

Thus, the correct increasing intervals are: $(-\infty, -1) \cup (0, 2)$

Decreasing Intervals

• The graph is decreasing between:
  • From x = -1 to x = 0 (between the local maximum at -1 and local minimum at 0).
  • From x = 2 onward.

Thus, the correct decreasing intervals are: $(-1, 0) \cup (2, \infty)$

Final Answer Summary:

Part (a):

• Local Maximum 1: $(-1, 1)$
• Local Maximum 2: $(2, 1)$
• Local Minimum: $(0, -1)$

Part (b):

• Increasing: $(-∞, -1) \cup (0, 2)$
• Decreasing: $(-1, 0) \cup (2, ∞)$

Let me know if you'd like further details or clarifications!

Here are 5 related questions to deepen understanding:

1. How can you identify local extrema using the first and second derivative tests?
2. What are the conditions for a function to be increasing or decreasing on an interval?
3. How do you determine points of inflection from a graph?
4. What role does concavity play in determining the shape of the graph?
5. How can we apply the derivative to find critical points of a function?

Tip: Always check the graph carefully at turning points, and ensure you consider nearby intervals for increasing or decreasing behavior!