https://file.aiquickdraw.com/mathbot/file/17261750976082gkty5wq.jpg

The photo shows a graph of a function \( f \) with specific points and arrows indicating local maxima and minima. Here’s the analysis based on the provided graph:

### Local Maximum
From the graph:
- There is a local maximum at \( x = \frac{\pi}{2} \)
- The local maximum value is 4.

### Local Minimum
From the graph:
- There is a local minimum at \( x = -\frac{\pi}{2} \)
- The local minimum value is -4.

**Answers to the homework:**
- **(a)** The number(s) at which \( f \) has a local maximum is \( x = \frac{\pi}{2} \).
  - **A.** The local maximum value is 4.
- **(b)** The number(s) at which \( f \) has a local minimum is \( x = -\frac{\pi}{2} \).
  - **A.** The local minimum value is -4.

Would you like further details or explanations on how to identify local extrema from a graph? Here are some related questions to consider:
1. How do you determine the nature of critical points using the first derivative test?
2. What role does the second derivative play in confirming whether a point is a maximum or minimum?
3. How can symmetry in a graph be used to infer other properties of the function?
4. Are there any real-world applications where identifying local maxima and minima would be particularly important?
5. How would you estimate the value of \( f(0) \) based on this graph?

**Tip:** When identifying local maxima or minima from a graph, it can be helpful to observe where the slope of the graph changes from positive to negative (for maxima) or negative to positive (for minima).

Using the given graph of the function f, find the following: (a) The numbers, if any, at which f has a local maximum. What are these local maximum values? (b) The numbers, if any, at which f has a local minimum. What are these local minimum values?

Learn how to find the local maxima and minima of a function by analyzing its graph. This problem involves understanding the concept of critical points, first and second derivative tests, and interpreting slope changes. Suitable for high school calculus students in Grades 11-12.

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