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} ).\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 ( (-infty,-3),(3, infty) ). it is concave downward on ( (-3,3) ).\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.)\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.

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} ).\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 ( (-infty,-3),(3, infty) ). it is concave downward on ( (-3,3) ).\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.)\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.

Answer

Explanation:

Step1: Find the second - derivative of (y = f(x)=\frac{3x^{2}+1}{x^{2}-9})

Use the quotient rule (y=\frac{u}{v}), where (u = 3x^{2}+1), (u^\prime=6x), (v=x^{2}-9), (v^\prime = 2x). First - derivative: (y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}=\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}}). Then, use the quotient rule again for (y^\prime=\frac{-56x}{(x^{2}-9)^{2}}), where (u=-56x), (u^\prime=-56), (v=(x^{2}-9)^{2}), (v^\prime = 2(x^{2}-9)\times2x = 4x(x^{2}-9)). Second - derivative: (y^{\prime\prime}=\frac{u^\prime v - uv^\prime}{v^{2}}=\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 - 3)^{3}(x + 3)^{3}}).

Step2: Analyze the sign of (y^{\prime\prime})

The numerator (168(x^{2}+3)>0) for all real (x) (since (x^{2}+3>0) for (x\in R)). For the denominator ((x - 3)^{3}(x + 3)^{3}):

  • When (x\in(-\infty,-3)), ((x - 3)^{3}(x + 3)^{3}<0), so (y^{\prime\prime}>0) (concave upward).
  • When (x\in(-3,3)), ((x - 3)^{3}(x + 3)^{3}>0), so (y^{\prime\prime}<0) (concave downward).
  • When (x\in(3,\infty)), ((x - 3)^{3}(x + 3)^{3}<0), so (y^{\prime\prime}>0) (concave upward).

Step3: Find inflection points

Inflection points occur where (y^{\prime\prime}) changes sign. But (y^{\prime\prime}) is not defined at (x=-3) and (x = 3) (since the function (y = f(x)) has vertical asymptotes at (x=-3) and (x = 3), and the domain of (y = f(x)) is (\mathbb{R}\setminus{-3,3})). So there are no inflection points in the domain of the function.

Answer:

For the concavity part: The function is concave upward on ((-\infty,-3),(3,\infty)). It is concave downward on ((-3,3)). For the inflection - point part: There are no inflection points.