Chapter 2.8, Problem 16E

a.

To determine

The intervals in which $f$ is increasing and decreasing.

Explanation of Solution

Given information:

The graph of the derivative $f\text{'}$ of a continuous function $f$ is shown.

  

Calculations:

Here the graph of the intersects the X- axis at $x=1,6\&8.$ .So the points are either maximum or minimum.

From the graph we can see that $f\text{'}$ goes from negative to positive in Y axis at $x=1$ and again goes negative Y axis at $x=6$ and again goes positive Y axis at $x=8$

Therefore it can be easily understood that, for $x=1\text{ to }x=2$ and $x=3\text{ to }x=6$ and $x=8\text{ to }x=\infty$ the function $f$ is increasing.

From the graph it can be easily understood that, for $x=-\infty \text{ to }x=1$ and $x=2\text{ to }x=3$ and $x=6\text{ to }x=8$ the function $f$ is decreasing.

b.

To determine

The value of $x$ for which $f$ has local maximum and minimum.

Explanation of Solution

Given information:

The graph of the derivative $f\text{'}$ of a continuous function $f$ is shown.

  

Calculations:

Here the graph of the intersects the X- axis at $x=1,6\&8.$ .So the points are either maximum or minimum.

From the graph we can see that $f\text{'}$ goes from negative to positive in Y axis at $x=1$ and again goes negative Y axis at $x=6$ and again goes positive Y axis at $x=8$ From the graph we can clearly see that,

For, $x=6$ , $f$ has local maximum, and

For, $x=8$ , $f$ has local minimum.

c.

To determine

The interval in which $f$ concave upward and downward.

Explanation of Solution

Given information:

The graph of the derivative $f\text{'}$ of a continuous function $f$ is shown.

  

Calculations:

Here the graph of the intersects the X- axis at $x=1,6\&8.$ .So the points are either maximum or minimum.

From the graph we can see that $f\text{'}$ goes from negative to positive in Y axis at $x=1$ and again goes negative Y axis at $x=6$ and again goes positive Y axis at $x=8$ .Therefore we can say that,

For $x=1\text{ to }x=2$ and $x=3\text{ to }x=6$ and $x=8\text{ to }x=\infty$ the function $f(x)$ concave upward.

For $x=-\infty \text{ to }x=1$ and $x=2\text{ to }x=3$ and $x=6\text{ to }x=8$ the function $f(x)$ concave downward.

d.

To determine

The point of inflection, in x coordinate, of the function $f$ .

Explanation of Solution

Given information:

The graph of the derivative $f\text{'}$ of a continuous function $f$ is shown.

  

Calculations:

Here the graph of the intersects the X- axis at $x=1,6\&8.$ .So the points are either maximum or minimum.

From the graph we can see that $f\text{'}$ goes from negative to positive in Y axis at $x=1$ and again goes negative Y axis at $x=6$ and again goes positive Y axis at $x=8$ .

The point of inflection, in x coordinate, of the function $f$ is $x=2\&7.$

d.

To determine

The graph of the function assuming $f(0)=0.$

Explanation of Solution

Given information:

The graph of the derivative $f\text{'}$ of a continuous function $f$ is shown.

  

Calculations:

Graph of the function is shown below.