Question

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=\frac{x + 2}{x - 2} ).

b. the function is increasing on. it is decreasing on.
c. the function is decreasing on ( (-infty,2),(2,infty) ). it is never increasing.
find 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.
a. there is a local minimum at ( x= ). there is no local maximum.
(type an integer or a decimal. use a comma to separate answers as needed.)
b. there is a local maximum at ( x= ). there is no local minimum.
(type an integer or a decimal. use a comma to separate answers as needed)
c. there is a local maximum at ( x= ) and there is a local minimum at ( x= ).
(type integers or decimals. use a comma to separate answers as needed.)
d. there are no local extrema.

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 \(u = x + 2\), \(u^\prime=1\), \(v=x - 2\), \(v^\prime = 1\).

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

Since \((x - 2)^{2}>0\) for all \(x
eq2\), and \(f^\prime(x)=\frac{-4}{(x - 2)^{2}}<0\) for all \(x
eq2\).

Answer:

D. There are no local extrema.