Chapter 1.2, Problem 25E

Solution

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

Local maxima at $\left(-1,4\right)$ and $\left(5,5\right)$ , local minimum at $\left(2,2\right)$ . The function increases on $(-\infty ,-1]$ , decreases on $\left[-1,2\right]$ , increases on $\left[2,5\right]$ and decreases on $[5,\infty )$ .

Given information: 

The graph

  

Calculation:

There are local maxima at $\left(-1,4\right)$ and $\left(5,5\right)$ because the function increases to those points and decreases after them.

There is a local minimum at $\left(2,2\right)$ because the function decreases to that point and increases after it.

The function increases on $(-\infty ,-1]$ , decreases on $\left[-1,2\right]$ , increases on $\left[2,5\right]$ and decreases on $[5,\infty )$ .

Therefore, Local maxima at $\left(-1,4\right)$ and $\left(5,5\right)$ , local minimum at $\left(2,2\right)$ . The function increases on $(-\infty ,-1]$ , decreases on $\left[-1,2\right]$ , increases on $\left[2,5\right]$ and decreases on $[5,\infty )$ .