Chapter 1.5, Problem 37E

To determine

The function is increasing, decreasing or constant.

Answer to Problem 37E

The function is increasing over the interval $\left(1,\infty \right)$ .

The function decreases over the interval $\left(-\infty ,-1\right)$ .

The function is constant over the interval $\left(-1,1\right)$ .

Explanation of Solution

Given information:

The given function

Calculation:

Let us find out the interval over which following function is constant, increasing or decreasing

  $f\left(x\right)=\left|x+1\right|+\left|x-1\right|$

Let $x_1$ and $x_2$ belong to any interval.

Without loss of generality, we assume $x_1<x_2$

If a function $f$ is constant, then $f\left(x_1\right)=f\left(x_2\right)$

If a function $f$ is increasing, then $f\left(x_1\right)<f\left(x_2\right)$

If a function $f$ is decreasing, then $f\left(x_1\right)>f\left(x_2\right)$

Using above definitions,

The function is increasing over the interval $\left(1,\infty \right)$ .

The function decreases over the interval $\left(-\infty ,-1\right)$ .

The function is constant over the interval $\left(-1,1\right)$ .