summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of (…

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=\frac{x + 2}{x - 2} ).\n\nb. the function is increasing on. it is decreasing on.\nc. the function is decreasing on ( (-infty,2),(2,infty) ). it is never increasing.\nfind the location of any local extrema of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. there is a local minimum at ( x= ). there is no local maximum.\n(type an integer or a decimal. use a comma to separate answers as needed.)\nb. there is a local maximum at ( x= ). there is no local minimum.\n(type an integer or a decimal. use a comma to separate answers as needed)\nc. there is a local maximum at ( x= ) and there is a local minimum at ( x= ).\n(type integers or decimals. use a comma to separate answers as needed.)\nd. there are no local extrema.

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=\frac{x + 2}{x - 2} ).\n\nb. the function is increasing on. it is decreasing on.\nc. the function is decreasing on ( (-infty,2),(2,infty) ). it is never increasing.\nfind the location of any local extrema of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. there is a local minimum at ( x= ). there is no local maximum.\n(type an integer or a decimal. use a comma to separate answers as needed.)\nb. there is a local maximum at ( x= ). there is no local minimum.\n(type an integer or a decimal. use a comma to separate answers as needed)\nc. there is a local maximum at ( x= ) and there is a local minimum at ( x= ).\n(type integers or decimals. use a comma to separate answers as needed.)\nd. there are no local extrema.

Answer

Explanation:

Step1: Find the derivative of (f(x))

Use the quotient rule ((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}). Here (u = x + 2), (u^\prime=1), (v=x - 2), (v^\prime = 1). [ \begin{align*} f^\prime(x)&=\frac{(x + 2)^\prime(x - 2)-(x + 2)(x - 2)^\prime}{(x - 2)^{2}}\ &=\frac{1\times(x - 2)-(x + 2)\times1}{(x - 2)^{2}}\ &=\frac{x-2-x - 2}{(x - 2)^{2}}\ &=\frac{-4}{(x - 2)^{2}} \end{align*} ]

Step2: Analyze the sign of (f^\prime(x))

Since ((x - 2)^{2}>0) for all (x\neq2), and (f^\prime(x)=\frac{-4}{(x - 2)^{2}}<0) for all (x\neq2).

Answer:

D. There are no local extrema.