Chapter 4.1, Problem 33E

To determine

To find : extreme values of the function and where they occur.

Answer to Problem 33E

Minimum $\left(2,-1\right)$ and maximum $\left(-2,7\right),\left(2,7\right)$

Explanation of Solution

Given information :

  $\left|3x-6\right|-\left|x-3\right|,-2\le x\le 5$

Formula needed :

For critical points, $\frac{dy}{dx}=0$ or $\frac{dy}{dx}$ is not defined.

The graph of the function in the domain is

  

Differentiate expression $\left|3x-6\right|-\left|x-3\right|$ with respect to $x$ .

  $\begin{array}{c}\frac{d}{dx}\left(\left|3x-6\right|-\left|x-3\right|\right)=\frac{d}{dx}\left(\left|3x-6\right|\right)-\frac{d}{dx}\left(\left|x-3\right|\right)\\ =\frac{\left|3x-6\right|}{3\left(x-2\right)}-\frac{\left|x-3\right|}{x-3}\end{array}$

Thus, the derivative of $\left|3x-6\right|-\left|x-3\right|$ never zero on $-2\le x\le 5$ .

Therefore, the derivative does not exist at $x=2$ and, $x=3$ .

The critical points are $x=2$ and, $x=3$ .

Examine the value of expression at critical points and end points.

  $\begin{array}{cc}x& \left|3x-6\right|-\left|x-3\right|\\ -2& 7\\ 2& -1\\ 3& 3\\ 5& 7\end{array}$

Hence, minimum occur at $x=2$ and maximum occur at end points.