Difference between revisions of "009A Sample Final 2, Problem 6"

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!Foundations:    
 
!Foundations:    
 
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|'''1.''' 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>
 
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&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>
 
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|'''2.''' 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>
 
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&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.
 
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
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|To find the absolute maximum and minimum of &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; on the interval &nbsp;<math style="vertical-align: -5px">[0,2],</math>
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|we need to find the critical points of &nbsp;<math style="vertical-align: -5px">f(x).</math>
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|Using the Quotient Rule, we have
 
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&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
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\displaystyle{f'(x)} & = & \displaystyle{\frac{(1+x)(1-x)'-(1-x)(1+x)'}{(1+x)^2}}\\
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&&\\
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& = & \displaystyle{\frac{(1+x)(-1)-(1-x)(1)}{(1+x)^2}}\\
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&&\\
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& = & \displaystyle{\frac{-2}{(1+x)^2}.}
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\end{array}</math>
 
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|We notice that &nbsp;<math style="vertical-align: -6px">f'(x)\ne 0</math>&nbsp; for any &nbsp;<math style="vertical-align: 0px">x.</math>
 
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|So, there are no critical points in the interval &nbsp;<math style="vertical-align: -5px">[0,2].</math>
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|Now, we have
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>f(0)=1,~f(2)=\frac{-1}{3}.</math>
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|Therefore, the absolute maximum value for &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is &nbsp;<math style="vertical-align: -1px">1</math>
 
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|and the absolute minimum value for &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is &nbsp;<math style="vertical-align: -15px">\frac{-1}{3}.</math>
 
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|&nbsp; &nbsp; &nbsp; &nbsp;The absolute maximum value for &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is &nbsp;<math style="vertical-align: -1px">1</math>&nbsp; and the absolute minimum value for &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is &nbsp;<math style="vertical-align: -15px">\frac{-1}{3}.</math>
 
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[[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

Revision as of 16:45, 7 March 2017

Find the absolute maximum and absolute minimum values of the function

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 [0,2].}

Foundations:  
1. To find the absolute maximum and minimum 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)}   on an 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 [a,b],}

        we need to compare the  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 y}   values of our critical points with  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(a)}   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(b).}

2. To find 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),}   we 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}   and solve 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 x.}

        Also, we include 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 x}   where  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.


Solution:

Step 1:  
To find the absolute maximum and minimum 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)}   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 [0,2],}
we need to find the critical points 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).}
Using the Quotient 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+x)(1-x)'-(1-x)(1+x)'}{(1+x)^2}}\\ &&\\ & = & \displaystyle{\frac{(1+x)(-1)-(1-x)(1)}{(1+x)^2}}\\ &&\\ & = & \displaystyle{\frac{-2}{(1+x)^2}.} \end{array}}

We notice that  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)\ne 0}   for any  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.}
So, there are no critical points in 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 [0,2].}
Step 2:  
Now, 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(0)=1,~f(2)=\frac{-1}{3}.}
Therefore, 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 1}
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 \frac{-1}{3}.}


Final Answer:  
       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 1}   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 \frac{-1}{3}.}

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