Chapter 1.3, Problem 41E

Solution

(a)

To find: The interval for which function is increasing or decreasing

Function is decreasing on $\left(-\infty ,2\right]$ and increasing on $\left[2,\infty \right)$ .

Given information:

The function is $s(x)=|x-2|$

The graph of the function is given below:

  

It can be observed from the graph that function is decreasing on $\left(-\infty ,2\right]$ and increasing on $\left[2,\infty \right)$ .Absolute minimum at x=2.

(b)

To find: The function is even.

Function is even

Given information:

The function is $sx)=|x-2|$

  $\begin{array}{l}s\left(-x\right)=|-x-2|\\ s\left(-x\right)=|x+2|\end{array}$

The function is even.

(C)

To find: The extrema of the function .

The function is absolute minimum at x=2

Given information:

The function is $s(x)=|x-2|$

Let $x<0$then $s(x)=-(x-2)=-x+2$

  $s\text{'}(x)=2$

  $s\text{'}\text{'}(x)=0$

The function has absolute minima at x=2

(d)

To find: The graph related to the 12 basic functions.

Function is related to basic absolute function.

Given information:

The function is $s(x)=|x-2|$

Function is related to basic absolute function.