(b) Find the​ value(s) of x at which f has a local minimum. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.

xequals=enter your response here

​(Type an exact​ answer, using 

piπ 

as needed. Use a comma to separate answers as​ needed.)

B.

There is no local minimum 

value.

### (b) Finding the Values of \( x \) at Which \( f \) Has a Local Minimum

#### Identifying the x-value of the local minimum:

From the graph description:
- The function falls to the point \((-\frac{\pi}{2}, -4)\) and then rises.

The function has a local minimum at \( x = -\frac{\pi}{2} \) because it falls to this point and then rises afterwards.

#### Answer:

A. \( x = -\frac{\pi}{2} \)

Would you like any further details or have any questions?

Here are five questions you can ask next:
1. How do you determine the local maxima and minima from a function's graph?
2. What is the first derivative test for finding local extrema?
3. Can you explain the second derivative test for concavity and points of inflection?
4. How do you find critical points of a function analytically?
5. What are global maximum and minimum values, and how do they differ from local extrema?

**Tip:** Always double-check the critical points by substituting them back into the function to confirm the exact values of the local maxima or minima.

Learn how to find the values of x at which a function has a local minimum using calculus. This problem involves identifying critical points and applying the first derivative test. Suitable for college-level students.

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