Based on the graph provided, let’s analyze the local extrema and intervals of the function. Local extrema occur at the points where the slope of the tangent is zero, marking a local maximum or minimum. Here's the step-by-step breakdown:

Local Extrema and Their Behavior:

1. Point A (-12, 1): Local maximum (increasing before, decreasing after).
2. Point B (-8, -1): Local minimum (decreasing before, increasing after).
3. Point C (-4, 1): Local maximum (increasing before, decreasing after).
4. Point D (0, -1): Local minimum (decreasing before, increasing after).
5. Point E (1.57, 1): Local maximum (increasing before, decreasing after).
6. Point F (4.71, -1): Local minimum (decreasing before, increasing after).
7. Point G (7.85, 1): Local maximum (increasing before, decreasing after).
8. Point H (11, -1): Local minimum (decreasing before, increasing after).

Intervals and Their Trends:

• Interval A-B: Decreasing (from 1 to -1).
• Interval B-C: Increasing (from -1 to 1).
• Interval C-D: Decreasing (from 1 to -1).
• Interval D-E: Increasing (from -1 to 1).
• Interval E-F: Decreasing (from 1 to -1).
• Interval F-G: Increasing (from -1 to 1).
• Interval G-H: Decreasing (from 1 to -1).

Difference Between Local Extrema and Global Extrema:

• Local Extrema: Points where the function has a peak (maximum) or valley (minimum) within a small neighborhood. For example, Point A is a local maximum because the value of the function decreases immediately to the left and right.
• Global Extrema: The highest or lowest points over the entire domain of the function. In the provided graph, no global extrema are explicitly visible since the function oscillates infinitely.

Would you like further clarification on any of these concepts? Let me know how I can assist further!

Follow-Up Questions for Deeper Understanding:

1. How can you identify global extrema in a bounded function?
2. What is the significance of derivatives in identifying these extrema?
3. How do intervals of increase or decrease relate to the behavior of extrema?
4. Can local extrema exist without having zero slopes? Why or why not?
5. How would this analysis change if the domain of the function were restricted?

Tip:

Always analyze the first derivative (slope) to identify intervals of increase/decrease, and the second derivative for concavity to confirm extrema type!