009A Sample Final 2, Problem 6 Detailed Solution

Find the absolute maximum and absolute minimum values of the function

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)=\frac{1-x}{1+x}}

on the interval  $[0,2].$

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Background Information:
1. 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  $f(b).$
2. 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.

Solution:

Step 1:
To find the absolute maximum and minimum of  $f(x)$  on the interval  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [0,2],}
we need to find the critical points of  $f(x).$
Using the Quotient Rule, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(x)\neq 0}   for any  $x.$
Also,  $f'(x)$  is undefined when  $x=-1.$  But, this is outside of our interval.
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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