Sketch the graph of *f* and use your sketch to find the absolute and local maximum and minimum values of *f*. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

*f*(*x*) = sin(*x*),    

−𝜋

2

 ≤ *x* ≤ 

𝜋

2

To sketch the graph of \( f(x) = \sin(x) \) for \( -\frac{\pi}{2} \le x \le \frac{\pi}{2} \), and find the absolute and local maximum and minimum values, follow these steps:

1. **Sketch the graph:**
   - The function \( \sin(x) \) is well-known and periodic with a range between -1 and 1.
   - The interval \( -\frac{\pi}{2} \le x \le \frac{\pi}{2} \) corresponds to the interval from the minimum point of \( \sin(x) \) at \( -\frac{\pi}{2} \) to the maximum point at \( \frac{\pi}{2} \).
   - At \( x = -\frac{\pi}{2} \), \( \sin\left(-\frac{\pi}{2}\right) = -1 \).
   - At \( x = 0 \), \( \sin(0) = 0 \).
   - At \( x = \frac{\pi}{2} \), \( \sin\left(\frac{\pi}{2}\right) = 1 \).

2. **Identify critical points:**
   - Since \( \sin(x) \) is continuous and differentiable within this interval, we consider endpoints and critical points where the derivative \( \cos(x) = 0 \).
   - However, \( \cos(x) = 0 \) does not have solutions within \( -\frac{\pi}{2} \le x \le \frac{\pi}{2} \).

3. **Find the local and absolute maxima and minima:**
   - Evaluate \( \sin(x) \) at the endpoints:
     \[
     f\left(-\frac{\pi}{2}\right) = \sin\left(-\frac{\pi}{2}\right) = -1
     \]
     \[
     f(0) = \sin(0) = 0
     \]
     \[
     f\left(\frac{\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) = 1
     \]
   - The absolute maximum value is \( 1 \) at \( x = \frac{\pi}{2} \).
   - The absolute minimum value is \( -1 \) at \( x = -\frac{\pi}{2} \).
   - There are no local maxima or minima within the open interval \( -\frac{\pi}{2} < x < \frac{\pi}{2} \), only at the endpoints.

Therefore, the absolute maximum value is \( 1 \) at \( x = \frac{\pi}{2} \), and the absolute minimum value is \( -1 \) at \( x = -\frac{\pi}{2} \). There are no other local extrema in the given interval.

To summarize:

- Absolute maximum: \( 1 \) at \( x = \frac{\pi}{2} \)
- Absolute minimum: \( -1 \) at \( x = -\frac{\pi}{2} \)
- Local maximum: None
- Local minimum: None

Would you like details or have any questions about this sketch and analysis?

Here are 8 related questions to expand your understanding:

1. How do you find critical points of a function?
2. What is the first derivative test for local extrema?
3. Can you find the derivative of \( f(x) = \sin(x) \)?
4. How do you determine the concavity of a function?
5. What is the second derivative test for local extrema?
6. How do you sketch the graph of \( f(x) = \cos(x) \) in a given interval?
7. What is the range of the function \( f(x) = \sin(x) \)?
8. How do you find the inflection points of a function?

**Tip:** When analyzing functions for extrema, always check the endpoints of the interval in addition to any critical points within the interval.

Learn how to sketch the graph of f(x) = sin(x) over the interval -π/2 ≤ x ≤ π/2 and find the absolute and local extrema. Explore detailed steps and calculations for identifying maximum and minimum values.

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