MathAdmin: Created page with "Let :: $g(x)=(2x^2-8x)^{\frac{2}{3}}$ (a) Find all critical points of $g$ ..."

2017-04-10T16:32:26Z

Created page with "Let ::<math>g(x)=(2x^2-8x)^{\frac{2}{3}}</math> (a) Find all critical points of <math style="vertical-align: -4px">g</math> ..."

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Let

::<math>g(x)=(2x^2-8x)^{\frac{2}{3}}</math>

(a) Find all critical points of  <math style="vertical-align: -4px">g</math>  over the  <math style="vertical-align: 0px">x</math>-interval  <math style="vertical-align: -5px">[0,8].</math>

(b) Find absolute maximum and absolute minimum of  <math style="vertical-align: -4px">g</math>  over  <math style="vertical-align: -5px">[0,8].</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|'''1.''' To find the critical points for  <math style="vertical-align: -5px">f(x),</math>  we set  <math style="vertical-align: -5px">f'(x)=0</math>  and solve for  <math style="vertical-align: -1px">x.</math>
|-
|
        Also, we include the values of  <math style="vertical-align: -1px">x</math>  where  <math style="vertical-align: -5px">f'(x)</math>  is undefined.
|-
|'''2.''' To find the absolute maximum and minimum of  <math style="vertical-align: -5px">f(x)</math>  on an interval  <math>[a,b],</math>
|-
|
        we need to compare the  <math style="vertical-align: -5px">y</math>  values of our critical points with  <math style="vertical-align: -5px">f(a)</math>  and  <math style="vertical-align: -5px">f(b).</math>
|}

'''Solution:'''

'''(a)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|To find the critical points, first we need to find  <math style="vertical-align: -5px">g'(x).</math>
|-
|Using the Chain Rule, we have
|-
|
       <math>\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}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|First, we note that  <math style="vertical-align: -5px">g'(x)</math>  is undefined when
|-
|       <math>3\sqrt[3]{2x^2-8x}=0.</math>
|-
|Solving for  <math style="vertical-align: -4px">x,</math>  we get
|-
|        <math>\begin{array}{rcl}
\displaystyle{0} & = & \displaystyle{2x^2-8x}\\
&&\\
& = & \displaystyle{x(2x-8).}
\end{array}</math>
|-
|Therefore,  <math style="vertical-align: -5px">g'(x)</math>  is undefined when  <math style="vertical-align: -4px">x=0,4.</math> 
|-
|Now, we need to set  <math style="vertical-align: -5px">g'(x)=0.</math>
|-
|So, we get
|-
|
       <math>8x-16=0.</math>
|-
|Solving, we get  <math style="vertical-align: 0px">x=2.</math>
|-
|Thus, the critical points for  <math style="vertical-align: -5px">f(x)</math>  are  <math style="vertical-align: -5px">(0,0),(2,4),(4,0).</math>
|}

'''(b)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|We need to compare the values of  <math style="vertical-align: -5px">g(x)</math>  at the critical points and at the endpoints of the interval.
|-
|Using the equation given, we have  <math style="vertical-align: -5px">g(0)=0</math>  and  <math style="vertical-align: -5px">g(8)=16.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Comparing the values in Step 1 with the critical points in (a), the absolute maximum value for  <math style="vertical-align: -5px">g(x)</math>  is  <math style="vertical-align: -1px">16</math>
|-
|and the absolute minimum value for  <math style="vertical-align: -5px">g(x)</math>  is  <math style="vertical-align: -1px">0.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|   '''(a)'''   <math>(0,0),(2,4),(4,0).</math>
|-
|   '''(b)'''   The absolute maximum value for  <math style="vertical-align: -5px">g(x)</math>  is  <math style="vertical-align: -1px">16</math>  and the absolute minimum value for  <math style="vertical-align: -5px">g(x)</math>  is  <math style="vertical-align: -1px">0.</math>
|}
[[009A_Sample_Final_3|'''Return to Sample Exam''']]

009A Sample Final 3, Problem 9 - Revision history

MathAdmin: Created page with "Let ::g(x)=(2x^2-8x)^{\frac{2}{3}} (a) Find all critical points of g ..."

MathAdmin: Created page with "Let :: $g(x)=(2x^2-8x)^{\frac{2}{3}}$ (a) Find all critical points of $g$ ..."