Difference between revisions of "009A Sample Final 3, Problem 9"

Revision as of 11:58, 7 March 2017

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 2:

(b)

Step 1:

Step 2:

Final Answer:
(a)
(b)

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