Question: In Problems 29–32, the graph of a function $f$ is given. Use the graph to find:
(a) The numbers, if any, at which $f$ has a local maximum. What are these local maxima?
(b) The numbers, if any, at which $f$ has a local minimum. What are these local minima?
30.
Local maximum at $x = 0$; local maximum value $2$.
Local minima at $x = -1$ and $x = 1$; local minimum value $0$.
Shape: a W-shaped graph with a peak at $(0, 2)$ and two valleys at $(-1, 0)$ and $(1, 0)$, positioned center.
31.
Local maximum at $x = \frac{\pi}{2}$; local maximum value $1$.
Local minimum at $x = -\frac{\pi}{2}$; local minimum value $-1$.
Shape: a sine-like wave crossing the x-axis at $-\pi$, $0$, and $\pi$, with a trough at $(-\frac{\pi}{2}, -1)$ and a crest at $(\frac{\pi}{2}, 1)$, positioned center-right.
32.
Local maximum at $x = 0$; local maximum value $1$.
Local minima at $x = -\pi$ and $x = \pi$; local minimum value $-1$.
Shape: a cosine-like wave with a peak at $(0, 1)$ and valleys at $(-\pi, -1)$ and $(\pi, -1)$, positioned bottom-right.
only do #30,31, and 32. DON"T DO #29!
1. **Problem Statement:**
We are given graphs of functions $f$ for problems 30, 31, and 32. We need to find the local maxima and minima points for each function.
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2. **Key Concepts:**
- A **local maximum** is a point where the function value is higher than all nearby points.
- A **local minimum** is a point where the function value is lower than all nearby points.
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### Problem 30
3. **Given:**
- Local maximum at $x=0$ with value $2$.
- Local minima at $x=-1$ and $x=1$ with value $0$.
4. **Explanation:**
- The graph is W-shaped.
- At $x=0$, the function reaches a peak (local max) of $2$.
- At $x=-1$ and $x=1$, the function reaches valleys (local mins) of $0$.
**Answer:**
- Local maxima: $x=0$, $f(0)=2$
- Local minima: $x=-1$, $f(-1)=0$ and $x=1$, $f(1)=0$
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### Problem 31
5. **Given:**
- Local maximum at $x=\frac{\pi}{2}$ with value $1$.
- Local minimum at $x=-\frac{\pi}{2}$ with value $-1$.
6. **Explanation:**
- The graph resembles a sine wave.
- The function crosses the x-axis at $-\pi$, $0$, and $\pi$.
- The trough (lowest point) is at $x=-\frac{\pi}{2}$ with value $-1$.
- The crest (highest point) is at $x=\frac{\pi}{2}$ with value $1$.
**Answer:**
- Local maxima: $x=\frac{\pi}{2}$, $f\left(\frac{\pi}{2}\right)=1$
- Local minima: $x=-\frac{\pi}{2}$, $f\left(-\frac{\pi}{2}\right)=-1$
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### Problem 32
7. **Given:**
- Local maximum at $x=0$ with value $1$.
- Local minima at $x=-\pi$ and $x=\pi$ with value $-1$.
8. **Explanation:**
- The graph resembles a cosine wave.
- The peak is at $x=0$ with value $1$.
- The valleys are at $x=-\pi$ and $x=\pi$ with value $-1$.
**Answer:**
- Local maxima: $x=0$, $f(0)=1$
- Local minima: $x=-\pi$, $f(-\pi)=-1$ and $x=\pi$, $f(\pi)=-1$