Chapter 1.2, Problem 26E

Solution

To identify: local minimum, a local maximum, or neither and function is decreasing and intervals on which it is increasing.

Local maxima at $\left(1,2\right)$ , local minimum at $\left(5,7\right)$ and is neither.

The function increases on $\left[1,5\right]$ , decreases on $(-\infty ,1]$ and decreases on $\left[5,\infty \right]$ .

Given information: 

The graph

  

Calculation:

Here,

  $\left(1,2\right)$ is a local minimum, because the graph descends until that point and rises after that point.

  $\left(3,3\right)$ is neither, because the graph rises until that point and continues rising after it.

  $\left(5,7\right)$ is a local maximum, because the graph rises until that point and then starts falling.

We see that the function decreases in the interval $(-\infty ,1]$ ;

increases in the interval $\left[1,5\right]$ ;

decreases in the interval $\left[5,\infty \right]$ .