Chapter 1.2, Problem 27E

Solution

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

The function $\left(-1,3\right)$ and $\left(3,3\right)$ are neither, $\left(1,5\right)$ is a local maximum, and $\left(5,1\right)$ is a local minimum.

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

Given information: 

The graph

  

Calculation:

Here,

  $\left(-1,3\right)$ and $\left(3,3\right)$ are neither because both of them keep moving in the same direction that they did before and after the graph passed through them.

  $\left(1,5\right)$ is a local maximum because the graph increases to that point and then decreases.

  $\left(5,1\right)$ is a local minimum because the graph decreases to that point and then increases.

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

Hence, $\left(-1,3\right)$ and $\left(3,3\right)$ are neither, $\left(1,5\right)$ is a local maximum, and $\left(5,1\right)$ is a local

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