https://file.aiquickdraw.com/mathbot/file/1726514698603tdpcrrne.jpg

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!

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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!

All the local maximum and minimum values of the function and the value of x at which each occurs. Also, identify the intervals where the function is increasing and decreasing.

Estimate the local maximum and minimum values of a given function, including identifying intervals where the function is increasing or decreasing. This problem involves applying first and second derivative tests to analyze extrema, suitable for high school or early college calculus students.

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