From the uploaded image, it looks like you are being asked to identify the local maxima, minima, and intervals of increase and decrease for a given function based on its graph.

Part (a) – Local Maxima and Minima

The task is to find:

1. Local Maxima: The highest points in small neighborhoods (peaks) of the graph.
2. Local Minima: The lowest points in small neighborhoods (valleys) of the graph.

For each of these, the x-coordinates where these maxima or minima occur need to be provided.

Part (b) – Intervals of Increase and Decrease

You need to determine:

• The intervals where the function is increasing (where the slope of the function is positive).
• The intervals where the function is decreasing (where the slope is negative).

Let me analyze the graph and extract the specific information for you.

Solution:

1. Local Maxima:

   • The graph shows a peak around $x = -2$.
   • There is another peak at approximately $x = 1$.
   • So, the local maxima occur at $x = -2$ and $x = 1$.

2. Local Minima:

   • The graph has a valley at around $x = 0.5$.
   • Thus, the local minima occurs at $x = 0.5$.

3. Intervals of Increase and Decrease:

   • Increasing: The graph is increasing (upward slope) between the intervals $(-\infty, -2)$ and $(0.5, \infty)$.
   • Decreasing: The graph is decreasing (downward slope) in the intervals $(-2, 0.5)$.

Summary:

• Local maxima: $x = -2, x = 1$
• Local minima: $x = 0.5$
• Intervals of increase: $(-\infty, -2), (0.5, \infty)$
• Intervals of decrease: $(-2, 0.5)$

Do you have any questions or would you like further details? Here are some related questions for additional practice:

1. How do you determine critical points on a graph?
2. What is the difference between absolute and local maxima/minima?
3. Can a function have more than one interval of increase or decrease?
4. How can you use derivatives to confirm local maxima or minima?
5. Why do we need to test endpoints when analyzing intervals of increase or decrease?

Tip: The first derivative of a function helps determine where the function is increasing or decreasing by showing where the slope is positive or negative.