Difference between revisions of "009A Sample Final 1, Problem 10"
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!Step 1: | !Step 1: | ||
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| − | | | + | |To find the critical point, first we need to find <math>f'(x)</math>. |
|- | |- | ||
| − | | | + | |Using the Product Rule, we have |
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| + | ::<math>\begin{array}{rcl} | ||
| + | \displaystyle{f'(x)} & = & \displaystyle{\frac{1}{3}x^{-\frac{2}{3}}(x-8)+x^{\frac{1}{3}}}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{\frac{x-8}{3x^{\frac{2}{3}}}+x^{\frac{1}{3}}}\\ | ||
| + | \end{array}</math> | ||
|} | |} | ||
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!Step 2: | !Step 2: | ||
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| − | | | + | |Notice <math>f'(x)</math> is undefined when <math>x=0</math>. |
| + | |- | ||
| + | |Now, we need to set <math>f'(x)=0</math>. | ||
| + | |- | ||
| + | |So, we get <math>-x^{\frac{1}{3}}=\frac{x-8}{3x^{\frac{2}{3}}}</math>. | ||
| + | |- | ||
| + | |We cross multiply to get <math>-3x=x-8</math>. | ||
|- | |- | ||
| − | | | + | |Solving, we get <math>x=2</math>. |
|- | |- | ||
| − | | | + | |Thus, the critical points for <math>f(x)</math> are <math>(0,0)</math> and <math>(2,2^{\frac{1}{3}}(-6))</math>. |
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
!Final Answer: | !Final Answer: | ||
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| − | |'''(a)''' | + | |'''(a)''' <math>(0,0)</math> and <math>(2,2^{\frac{1}{3}}(-6))</math> |
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|'''(b)''' | |'''(b)''' | ||
|} | |} | ||
[[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']] | [[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 16:59, 14 February 2016
Consider the following continuous function:
defined on the closed, bounded interval .
![{\displaystyle [-8,8]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4becc4d98ef5cf1d8c91c1b84fd9af8d5eeb50f1)
a) Find all the critical points for .
b) Determine the absolute maximum and absolute minimum values for on the interval .
| Foundations: |
|---|
Solution:
(a)
| Step 1: |
|---|
| To find the critical point, first we need to find . |
| Using the Product Rule, we have |
|
|
| Step 2: |
|---|
| Notice 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)} is undefined when 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=0} . |
| 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 f'(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 -x^{\frac{1}{3}}=\frac{x-8}{3x^{\frac{2}{3}}}} . |
| We cross multiply to 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 -3x=x-8} . |
| 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)} 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 (2,2^{\frac{1}{3}}(-6))} . |
(b)
| Step 1: |
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| Step 2: |
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| Final Answer: |
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| (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)} 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 (2,2^{\frac{1}{3}}(-6))} |
| (b) |