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} ).\nfind the location of any inflection points of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. there is an inflection point at ( x = ).\n(type an integer or a decimal. use a comma to separate answers as needed.)\nb. there are no inflection points.\nwhich graph below shows ( f(x) )?

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=\frac{x + 2}{x - 2} ).\nfind the location of any inflection points of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. there is an inflection point at ( x = ).\n(type an integer or a decimal. use a comma to separate answers as needed.)\nb. there are no inflection points.\nwhich graph below shows ( f(x) )?

Answer

Explanation:

Step1: Find the first derivative

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: Find the second derivative

Use the quotient rule again. Let (u=-4), (u^\prime = 0), (v=(x - 2)^{2}), (v^\prime=2(x - 2)) [ \begin{align*} f^{\prime\prime}(x)&=\frac{0\times(x - 2)^{2}-(-4)\times2(x - 2)}{(x - 2)^{4}}\ &=\frac{8(x - 2)}{(x - 2)^{4}}\ &=\frac{8}{(x - 2)^{3}} \end{align*} ]

Step3: Find inflection points

Set (f^{\prime\prime}(x)=0), (\frac{8}{(x - 2)^{3}}=0). There is no solution for (x) since the numerator (8\neq0).

Answer:

B. There are no inflection points.