009A Sample Final 3, Problem 9

Let

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

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

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

Foundations:
1. To find the critical points for  $f(x),$  we set  $f'(x)=0$  and solve for  $x.$
Also, we include the values of  $x$  where  $f'(x)$  is undefined.
2. To find the absolute maximum and minimum of  $f(x)$  on an interval  $[a,b],$
we need to compare the  $y$  values of our critical points with  $f(a)$  and  $f(b).$

Solution:

(a)

Step 1:
To find the critical points, first we need to find  $g'(x).$
Using the Chain Rule, we have
${\begin{array}{rcl}\displaystyle {g'(x)}&=&\displaystyle {{\frac {2}{3}}(2x^{2}-8x)^{-{\frac {1}{3}}}(2x^{2}-8x)'}\\&&\\&=&\displaystyle {{\frac {2}{3}}(2x^{2}-8x)^{-{\frac {1}{3}}}(4x-8)}\\&&\\&=&\displaystyle {{\frac {8x-16}{3{\sqrt[{3}]{2x^{2}-8x}}}}.}\end{array}}$

Step 2:
First, we note that  $g'(x)$  is undefined when
$3{\sqrt[{3}]{2x^{2}-8x}}=0.$
Solving for  $x,$  we get
${\begin{array}{rcl}\displaystyle {0}&=&\displaystyle {2x^{2}-8x}\\&&\\&=&\displaystyle {x(2x-8).}\end{array}}$
Therefore,  $g'(x)$  is undefined when  $x=0,4.$
Now, we need to set  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g'(x)=0.}
So, we get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8x-16=0.}
Solving, we get  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=2.}
Thus, the critical points for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)}   are  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,0),(2,4),(4,0).}

(b)

Step 1:
We need to compare the values of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)}   at the critical points and at the endpoints of the interval.
Using the equation given, we have  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(0)=0}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(8)=16.}

Step 2:
Comparing the values in Step 1 with the critical points in (a), the absolute maximum value for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)}   is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 16}
and the absolute minimum value for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)}   is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.}

Final Answer:
(a)   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,0),(2,4),(4,0).}
(b)   The absolute maximum value for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)}   is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 16}   and the absolute minimum value for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)}   is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.}

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