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?
x
y
negative pipinegative StartFraction pi Over 2 EndFraction
StartFraction pi Over 2 EndFraction
left parenthesis StartFraction pi Over 2 EndFraction comma 4 right parenthesis
left parenthesis negative StartFraction pi Over 2 EndFraction comma negative 4 right parenthesis

A coordinate system has a horizontal x-axis and a vertical y-axis. From left to right, a curve falls in quadrant 2 through the plotted point (negative pi, 0) to the plotted point (StartFraction negative pi Over 2 EndFraction, negative 4), rises through the origin to the plotted point (StartFraction pi Over 2 EndFraction, 4), and falls passing through the plotted point (pi, 0).
Question content area bottom
Part 1
​(a)  Find the​ value(s) of x at which f has a local maximum. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
xequals

StartFraction pi Over 2 EndFraction
​(Type an exact​ answer, using pi as needed. Use a comma to separate answers as​ needed.)
Your answer is correct.B.
There is no local maximum value.
Part 2
Find the local maximum. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The local maximum is

enter your response here.
​(Type an exact​ answer, using pi as needed. Use a comma to separate answers as​ needed.)
B.
There is no local maximum.

To solve the given problem, we need to analyze the graph of the function \( f \) and identify the local maximum and minimum points. The graph description and the given points will help us determine these.

### (a) Local Maximum

#### Finding the x-value of the local maximum:
From the graph description:
- The function falls from \((-\pi, 0)\) to \((-\frac{\pi}{2}, -4)\).
- Then it rises to \((0, 0)\) and continues to \((\frac{\pi}{2}, 4)\).
- Finally, it falls again passing through \((\pi, 0)\).

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

**Answer:**
\( x = \frac{\pi}{2} \)

#### Finding the value of the local maximum:
At \( x = \frac{\pi}{2} \), the function value is \( f\left(\frac{\pi}{2}\right) = 4 \).

**Answer:**
The local maximum value is \( 4 \).

### (b) Local Minimum

#### Finding the x-value of the local minimum:
From the graph description:
- The function falls to \((-\frac{\pi}{2}, -4)\) 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:**
\( x = -\frac{\pi}{2} \)

#### Finding the value of the local minimum:
At \( x = -\frac{\pi}{2} \), the function value is \( f\left(-\frac{\pi}{2}\right) = -4 \).

**Answer:**
The local minimum value is \( -4 \).

### Summary:
(a) The function \( f \) has a local maximum at \( x = \frac{\pi}{2} \) with the value \( 4 \).

(b) The function \( f \) has a local minimum at \( x = -\frac{\pi}{2} \) with the value \( -4 \).

Do you want more 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:** To find local maxima and minima of a function, look for points where the derivative changes sign. A positive to negative change indicates a local maximum, and a negative to positive change indicates a local minimum.

Learn how to find local maximum and minimum values of a function using its graph. This problem involves analyzing the graph of a function to identify points of local maximum and minimum. Suitable for high school students studying functions and graph analysis.

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