Chapter 4.3, Problem 6E

(a)

To determine

To find:The interval where $f$ is increasing.

Explanation of Solution

The graph of $f\left(x\right)$ increase when $f^{\prime }\left(x\right)>0$ .

It is visible from the graph that,

  $f^{\prime }\left(x\right)>0\text{when}x\in \left(2,4\right)\cup \left(6,9\right)$

Therefore, the function increases in the interval when $x\in \left(2,4\right)\cup \left(6,9\right)$

(b)

To determine

To find :The value of $x$ for which $f$ have a local maxima and minima.

Explanation of Solution

The function $f$ has a local maxima or local minima when its derivative crosses the $x$ axis

It is visible from the graph that $f^{\prime }\left(x\right)$ crosses the $x$ axis at $x=2,x=4,x=6$

  $f^{\prime }\left(x\right)>0\text{when}x\in \left(2,4\right)\cup \left(6,9\right)$

Therefore, the value of $x$ are $2,4,6$

(c)

To determine

To find :The interval on which $f$ is concave upward or downward.

Explanation of Solution

The graph is concave up when $f^{″}\left(x\right)>0$ and concave downward when $f^{″}\left(x\right)<0$ .

The derivative of $f^{\prime }\left(x\right)$ is positive when $f^{\prime }\left(x\right)$ is increasing.

The derivative of $f^{\prime }\left(x\right)$ is negative when $f^{\prime }\left(x\right)$ is decreasing.

It is visible from the graph that the graph is concave up

  $x\in \left(1,3\right)\cup \left(5,7\right)\cup \left(8,9\right)$

The graph is concave down when,

  $x\in \left(0,1\right)\cup \left(3,5\right)\cup \left(7,8\right)$

(d)

To determine

To find :The $x$ coordinates of the inflection points.

Explanation of Solution

The inflection point occur when the graph changes its concavity.

The graph is concave up at $x\in \left(1,3\right)\cup \left(5,7\right)\cup \left(8,9\right)$ .

The graph is concave down at $x\in \left(0,1\right)\cup \left(3,5\right)\cup \left(7,8\right)$ .

Therefore, the inflection point occurs at $x=1,3,5,7,8$