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).