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{3 x^{2}+1}{x^{2}-9} ).\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.\nfind the intervals where ( f(x) ) is concave upward or downward. select the correct choice below and fill in the answer box(es) to complete your choice.\na. the function is concave upward on . it is never concave downward.\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function is concave upward on . it is concave downward on .\n(type your answers in interval notation. use a comma to separate answers as needed.)\nc. the function is concave downward on . it is never concave upward.\n(type your answer in interval notation. use a comma to separate answers as needed.)

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=\frac{3 x^{2}+1}{x^{2}-9} ).\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.\nfind the intervals where ( f(x) ) is concave upward or downward. select the correct choice below and fill in the answer box(es) to complete your choice.\na. the function is concave upward on . it is never concave downward.\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function is concave upward on . it is concave downward on .\n(type your answers in interval notation. use a comma to separate answers as needed.)\nc. the function is concave downward on . it is never concave upward.\n(type your answer in interval notation. use a comma to separate answers as needed.)

Answer

Explanation:

Step1: Find the first derivative

Use the quotient rule ((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}). Let (u = 3x^{2}+1), (u^\prime=6x); (v=x^{2}-9), (v^\prime = 2x). Then (f^\prime(x)=\frac{6x(x^{2}-9)-2x(3x^{2}+1)}{(x^{2}-9)^{2}}=\frac{6x^{3}-54x - 6x^{3}-2x}{(x^{2}-9)^{2}}=\frac{- 56x}{(x^{2}-9)^{2}}). Set (f^\prime(x)=0), then (-56x = 0), so (x = 0).

Step2: Analyze the sign of the first derivative

When (x\lt0), (f^\prime(x)\gt0); when (x\gt0), (f^\prime(x)\lt0). So (x = 0) is a local maximum.

Step3: Find the second derivative

Use the quotient rule again. Let (u=-56x), (u^\prime=-56); (v=(x^{2}-9)^{2}), (v^\prime = 2(x^{2}-9)\times2x=4x(x^{2}-9)). Then (f^{\prime\prime}(x)=\frac{-56(x^{2}-9)^{2}+56x\times4x(x^{2}-9)}{(x^{2}-9)^{4}}=\frac{-56(x^{2}-9)+224x^{2}}{(x^{2}-9)^{3}}=\frac{-56x^{2}+504 + 224x^{2}}{(x^{2}-9)^{3}}=\frac{168x^{2}+504}{(x^{2}-9)^{3}}=\frac{168(x^{2}+3)}{(x^{2}-9)^{3}}).

Step4: Analyze the concavity

Set (f^{\prime\prime}(x)=0), since (x^{2}+3\gt0) for all real (x), the numerator (168(x^{2}+3)\gt0). The denominator ((x^{2}-9)^{3}\gt0) when (x^{2}-9\gt0) (i.e., (x\lt - 3) or (x\gt3)) and ((x^{2}-9)^{3}\lt0) when (x^{2}-9\lt0) (i.e., (-3\lt x\lt3)). When (x\lt - 3) or (x\gt3), (f^{\prime\prime}(x)\gt0) (concave upward); when (-3\lt x\lt3), (f^{\prime\prime}(x)\lt0) (concave downward).

Answer:

C. There is a local maximum at (x = 0) and there is no local minimum. B. The function is concave upward on ((-\infty,-3),(3,\infty)). It is concave downward on ((-3,3)).