Let's break down the information and analyze the graph step by step to answer each part of the question.

Part 1: Identify the nature of the point $(-2, -1)$

The graph describes two curves. One curve rises to a maximum and then falls, and the maximum occurs at the point $(-2, -1)$. Since the function reaches its highest point at this coordinate before decreasing, this is a local maximum.

• Answer: A. a local maximum

Part 2: Identify the nature of the point $(1, -10)$

From the graph, the curve falls to this point and then rises again. This point is the lowest point for this section of the curve, making it a local minimum.

• Answer: A. a local minimum

Part 3: Identify the nature of the point $(4, -21)$

This is described as the lowest point on the second curve, which then rises afterward. Since the function reaches its minimum here and increases after that, this is a local minimum.

• Answer: C. a local minimum

Part 4: Intervals where the function is increasing

The function is increasing where the slope of the curve is positive. Based on the description:

• The first curve is increasing before reaching the local maximum at $(-2, -1)$.
• The second curve is increasing after reaching the local minimum at $(4, -21)$.

Thus, the function is increasing on the intervals $(-\infty, -2)$ and $[4, \infty)$.

• Answer: D. $(- \infty, -2]$ and $[4, \infty)$

Part 5: Intervals where the function is decreasing

The function is decreasing where the slope of the curve is negative. From the graph description:

• The first curve is decreasing after reaching the local maximum at $(-2, -1)$ and continues until the local minimum at $(1, -10)$.
• The function is also decreasing from $(-2, 1)$ to $(4, -21)$.

Thus, the function is decreasing on the interval $[ -2, 4 ]$.

• Answer: C. $[-2, 4]$

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Summary of Answers:

1. Part 1: A. Local maximum
2. Part 2: A. Local minimum
3. Part 3: C. Local minimum
4. Part 4: D. $(- \infty, -2]$ and $[4, \infty)$
5. Part 5: C. $[-2, 4]$

Questions to explore further:

1. How does the derivative of the function relate to increasing and decreasing intervals?
2. What is the significance of a local extremum in the context of a real-world problem?
3. How would you use the second derivative to classify the nature of the critical points?
4. How would the graph change if we shifted the function vertically?
5. What happens if we only consider one curve independently? Would the conclusions about extrema and intervals change?

Tip: To find local extrema, it's helpful to look for where the derivative is zero or undefined, then determine whether the function changes direction at those points.