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?
29.
30.
31.
32.
29: graph with a local maximum at $x = 0$, $f(0) = 3$, and local minima at $x = -2$ and $x = 2$, with $f(-2) = 0$ and $f(2) = 0$. position_hint = top-left
30: graph with a local maximum at $x = 0$, $f(0) = 2$, and local minima at $x = -1$ and $x = 1$, with $f(-1) = 0$ and $f(1) = 0$. position_hint = top-left
31: graph with a local maximum at $x = \frac{\pi}{2}$, $f\left(\frac{\pi}{2}\right) = 1$, and a local minimum at $x = -\frac{\pi}{2}$, $f\left(-\frac{\pi}{2}\right) = -1$. position_hint = top-right
32: graph with a local maximum at $x = 0$, $f(0) = 1$, and local minima at $x = -\pi$ and $x = \pi$, with $f(-\pi) = -1$ and $f(\pi) = -1$. position_hint = top-right
1. **State the problem:** We are given graphs of functions $f$ for Problems 29 to 32. For each, we need to find:
(a) The $x$-values where $f$ has local maxima and the corresponding maximum values.
(b) The $x$-values where $f$ has local minima and the corresponding minimum values.
2. **Recall definitions:**
- A **local maximum** at $x = a$ means $f(a)$ is greater than or equal to $f(x)$ for all $x$ near $a$.
- A **local minimum** at $x = b$ means $f(b)$ is less than or equal to $f(x)$ for all $x$ near $b$.
3. **Analyze each problem:**
**Problem 29:**
- Local maximum at $x=0$ with $f(0)=3$.
- Local minima at $x=-2$ and $x=2$ with $f(-2)=0$ and $f(2)=0$.
**Problem 30:**
- Local maximum at $x=0$ with $f(0)=2$.
- Local minima at $x=-1$ and $x=1$ with $f(-1)=0$ and $f(1)=0$.
**Problem 31:**
- Local maximum at $x=\frac{\pi}{2}$ with $f\left(\frac{\pi}{2}\right)=1$.
- Local minimum at $x=-\frac{\pi}{2}$ with $f\left(-\frac{\pi}{2}\right)=-1$.
**Problem 32:**
- Local maximum at $x=0$ with $f(0)=1$.
- Local minima at $x=-\pi$ and $x=\pi$ with $f(-\pi)=-1$ and $f(\pi)=-1$.
4. **Summary of answers:**
- Problem 29:
- Local maxima: $x=0$, $f(0)=3$
- Local minima: $x=-2$, $f(-2)=0$ and $x=2$, $f(2)=0$
- Problem 30:
- Local maxima: $x=0$, $f(0)=2$
- Local minima: $x=-1$, $f(-1)=0$ and $x=1$, $f(1)=0$
- Problem 31:
- Local maxima: $x=\frac{\pi}{2}$, $f\left(\frac{\pi}{2}\right)=1$
- Local minima: $x=-\frac{\pi}{2}$, $f\left(-\frac{\pi}{2}\right)=-1$
- Problem 32:
- Local maxima: $x=0$, $f(0)=1$
- Local minima: $x=-\pi$, $f(-\pi)=-1$ and $x=\pi$, $f(\pi)=-1$