Chapter 2.3, Problem 19AYU

In Problems 13-24, use the graph of the function $f$ given.

21. List the number(s) at which $f$ has a local maximum. What are the local maximum values?

To determine

All the local maximum of $f$ with corresponding values using the graph.

Answer to Problem 19AYU

The local maximum points are $\left(-2,6\right)\text{ and }\left(2,10\right)$.

Explanation of Solution

Given:

It is asked to find all the local maximum of $f$ with corresponding values using the graph.

Graph:

Interpretation:

By the definition of local maximum, “Let $f$ be a function defined on some interval $I$.

A function $f$ has a local maximum at $c$ if there is an open interval $I$ containing $c$ so that, for all $x$ in this open interval, we have $f\left(x\right)\le f\left(c\right)$. We call $f\left(c\right)$ a local maximum value of $f$”, It can be directly concluded from the graph and the definition that the curve has local maximum points at

$x=-2$ and $x=2$

The values of the local maximum at $x=-2$ is 6 and the local maximum at $x=2$ is 10.

Therefore, the local maximum points are $\left(-2,6\right)$ and $\left(2,10\right)$.