Chapter 2.8, Problem 1E

a.

To determine

To find:The interval of the given graph in which the function is increasing or decreasing.

Answer to Problem 1E

Increasing: $\left(0,1\right)\cup \left(4,5\right)$

Decreasing: $\left(1,4\right)$

Explanation of Solution

Given:

  

When the derivative of a differentiable function is positive, then the function is increasing on that interval.

It observes that $f^{\text{'}}\left(x\right)$ is positive on the intervals $\left(0,1\right)$

  $\text{and }\left(4,5\right)$ .

Therefore, it is increasing on the same intervals.

When the derivative of a differentiable function is negative, then the function is decreasing on that interval.

It observes that $f^{\text{'}}\left(x\right)$ is negative on the interval $\left(1,4\right)$ .

Therefore, it is decreasing on the same intervals.

b.

To determine

To find: The value of $x$ which have local maximum and local minimum of given graph.

Answer to Problem 1E

Local maximum: $1$ .

Local minimum: $4$ .

Explanation of Solution

Given:

  

$f\left(x\right)$ is concave down on $\left(0,2\right)$ since $f^{\text{'}\text{'}}\left(x\right)<0$ and concave up on since $f^{\text{'}\text{'}}\left(x\right)>0$ .

1 and 4 are the local maxima/minima sine they both hit $y=0$ on the $f^{\text{'}}\left(x\right)$ graph.

1 is a local maximum because it is on a concave up region, and 1 is a local minimum and 4 is a local minimum because it’s on a concave up region.

c.

To determine

To sketch: A possible graph of the given function where $f\left(0\right)=0$ .

Explanation of Solution

Given:

  

Graph the starting point of function at $\left(0,0\right)$ it is given.

  $f^{\text{'}}\left(x\right)$ is starts positive.

  $f^{\text{'}}\left(x\right)$ crosses the x- axis therefore, $f\left(x\right)$ is flat in middle.

  $f^{\text{'}}\left(x\right)$ starts going up and where it rosses the x-axis is flat.