009A Sample Final 3, Problem 9

Let

${\displaystyle g(x)=(2x^{2}-8x)^{\frac {2}{3}}}$

(a) Find all critical points of  ${\displaystyle g}$  over the  ${\displaystyle x}$-interval  ${\displaystyle [0,8].}$

(b) Find absolute maximum and absolute minimum of  ${\displaystyle g}$  over  ${\displaystyle [0,8].}$

Foundations:
1. To find the critical points for  ${\displaystyle f(x),}$  we set  ${\displaystyle f'(x)=0}$  and solve for  ${\displaystyle x.}$

Also, we include the values of  ${\displaystyle x}$  where  ${\displaystyle f'(x)}$  is undefined.

2. To find the absolute maximum and minimum of  ${\displaystyle f(x)}$  on an interval  ${\displaystyle [a,b],}$

we need to compare the  ${\displaystyle y}$  values of our critical points with  ${\displaystyle f(a)}$  and  ${\displaystyle f(b).}$

Solution:

(a)

Step 1:
To find the critical points, first we need to find  ${\displaystyle g'(x).}$
Using the Chain Rule, we have

${\displaystyle {\begin{array}{rcl}\displaystyle {g'(x)}&=&\displaystyle {{\frac {2}{3}}(2x^{2}-8x)^{-{\frac {1}{3}}}(2x^{2}-8x)'}\\&&\\&=&\displaystyle {{\frac {2}{3}}(2x^{2}-8x)^{-{\frac {1}{3}}}(4x-8)}\\&&\\&=&\displaystyle {{\frac {8x-16}{3{\sqrt[{3}]{2x^{2}-8x}}}}.}\end{array}}}$

Step 2:
First, we note that  ${\displaystyle g'(x)}$  is undefined when
${\displaystyle 3{\sqrt[{3}]{2x^{2}-8x}}=0.}$
Solving for  ${\displaystyle x,}$  we get
${\displaystyle {\begin{array}{rcl}\displaystyle {0}&=&\displaystyle {2x^{2}-8x}\\&&\\&=&\displaystyle {x(2x-8).}\end{array}}}$
Therefore,  ${\displaystyle g'(x)}$  is undefined when  ${\displaystyle x=0,4.}$
Now, we need to set  ${\displaystyle g'(x)=0.}$
So, we get

${\displaystyle 8x-16=0.}$

Solving, we get  ${\displaystyle x=2.}$
Thus, the critical points for  ${\displaystyle f(x)}$  are  ${\displaystyle (0,0),(2,4),(4,0).}$

(b)

Step 1:
We need to compare the values of  ${\displaystyle g(x)}$  at the critical points and at the endpoints of the interval.
Using the equation given, we have  ${\displaystyle g(0)=0}$  and  ${\displaystyle g(8)=16.}$
Step 2:
Comparing the values in Step 1 with the critical points in (a), the absolute maximum value for  ${\displaystyle g(x)}$  is  ${\displaystyle 16}$
and the absolute minimum value for  ${\displaystyle g(x)}$  is  ${\displaystyle 0.}$

(a)   ${\displaystyle (0,0),(2,4),(4,0).}$
(b)   The absolute maximum value for  ${\displaystyle g(x)}$  is  ${\displaystyle 16}$  and the absolute minimum value for  ${\displaystyle g(x)}$  is  ${\displaystyle 0.}$