Difference between revisions of "009A Sample Final 1, Problem 10"

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!Step 1:    
 
!Step 1:    
 
|-
 
|-
−
|To find the critical point, first we need to find <math>f'(x)</math>.
+
|To find the critical point, first we need to find <math style="vertical-align: -5px">f'(x)</math>.
 
|-
 
|-
 
|Using the Product Rule, we have
 
|Using the Product Rule, we have
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
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|Notice <math>f'(x)</math> is undefined when <math>x=0</math>.  
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|Notice <math style="vertical-align: -5px">f'(x)</math> is undefined when <math style="vertical-align: -1px">x=0</math>.  
 
|-
 
|-
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|Now, we need to set <math>f'(x)=0</math>.
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|Now, we need to set <math style="vertical-align: -5px">f'(x)=0</math>.
 
|-
 
|-
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|So, we get <math>-x^{\frac{1}{3}}=\frac{x-8}{3x^{\frac{2}{3}}}</math>.
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|So, we get  
 
|-
 
|-
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|We cross multiply to get <math>-3x=x-8</math>.
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|
 +
::<math>-x^{\frac{1}{3}}=\frac{x-8}{3x^{\frac{2}{3}}}</math>.
 +
|-
 +
|We cross multiply to get <math style="vertical-align: 1px">-3x=x-8</math>.
 
|-
 
|-
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|Solving, we get <math>x=2</math>.
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|Solving, we get <math style="vertical-align: -1px">x=2</math>.
 
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|-
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|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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|Thus, the critical points for <math style="vertical-align: -5px">f(x)</math> are <math style="vertical-align: -4px">(0,0)</math> and <math style="vertical-align: -4px">(2,2^{\frac{1}{3}}(-6))</math>.
 
|}
 
|}
  
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
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|We need to compare the values of <math>f(x)</math> at the critical points and at the endpoints of the interval.  
+
|We need to compare the values of <math style="vertical-align: -5px">f(x)</math> at the critical points and at the endpoints of the interval.  
 
|-
 
|-
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|Using the equation given, we have <math>f(-8)=32</math> and <math>f(8)=0</math>.
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|Using the equation given, we have <math style="vertical-align: -5px">f(-8)=32</math> and <math style="vertical-align: -5px">f(8)=0</math>.
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
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|Comparing the values in Step 1 with the critical points in '''(a)''', the absolute maximum value for <math>f(x)</math> is 32  
+
|Comparing the values in Step 1 with the critical points in '''(a)''', the absolute maximum value for <math style="vertical-align: -5px">f(x)</math> is <math style="vertical-align: -1px">32</math>
 
|-
 
|-
−
|and the absolute minimum value for <math>f(x)</math> is <math>2^{\frac{1}{3}}(-6)</math>.
+
|and the absolute minimum value for <math style="vertical-align: -5px">f(x)</math> is <math style="vertical-align: -5px">2^{\frac{1}{3}}(-6)</math>.
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
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|'''(a)''' <math>(0,0)</math> and <math>(2,2^{\frac{1}{3}}(-6))</math>
+
|'''(a)''' <math style="vertical-align: -4px">(0,0)</math> and <math style="vertical-align: -4px">(2,2^{\frac{1}{3}}(-6))</math>
 
|-
 
|-
−
|'''(b)''' The absolute minimum value for <math>f(x)</math> is <math>2^{\frac{1}{3}}(-6)</math>
+
|'''(b)''' The absolute minimum value for <math style="vertical-align: -5px">f(x)</math> is <math style="vertical-align: -5px">2^{\frac{1}{3}}(-6)</math>.
 
|}
 
|}
 
[[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]

Revision as of 16:58, 22 February 2016

Consider the following continuous function:

defined on the closed, bounded interval .

a) Find all 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)} .

b) Determine the absolute maximum and absolute minimum values 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)} on the interval 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 [-8,8]} .

Foundations:  

Solution:

(a)

Step 1:  
To find the critical point, first we need to find 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)} .
Using the Product Rule, 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 \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}}
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:  
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 f(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 f(-8)=32} 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 f(8)=0} .
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 f(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 32}
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 f(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 2^{\frac{1}{3}}(-6)} .
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)} 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) 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 f(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 2^{\frac{1}{3}}(-6)} .

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