Chapter 2.2, Problem 23E

$23.$

Refer to figure $21,$

Looking at the graph $f\text{'}\left(x\right),$ determine whether $f\left(x\right)$ is increasing or decreasing at $x=9$. Look at the graph of $f\left(x\right)$ to confirm your answer.

Looking at the value of $f\text{'}\left(x\right)$ for $1\le x<2$

and $2<x\le 3,$

explain why the graph of $f\left(x\right)$

must have a relative maximum at $x=2$. What are the coordinates of the relative maximum point?

Looking at the value of $f\left(x\right)$ for $x$ close to $10,$

explain why the graph of $f\left(x\right)$

has a relative minimum at $x=10$.

Looking at the graph of $f\text{'}\text{'}\left(x\right),$ determine whether $f\left(x\right)$ is concave up or concave down at $x=2$. Look at the graph of $f\left(x\right)$ to confirm your answer.

Looking at the graph of $f\text{'}\text{'}\left(x\right),$ determine where $f\left(x\right)$ has an inflection point. Look at the graph of $f\left(x\right)$ to confirm your answer. What are the coordinates of the inflection point?

Find the $x-$

coordinate of the point on the graph at $f\left(x\right)$ is increasing at a rate of $6$

units per unit change in $x$.