Question

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} ).
a. the function is concave upward on. it is never concave downward.
(type your answer in interval notation. use a comma to separate answers as needed.)
b. the function is concave upward on ( (-infty,-3),(3, infty) ). it is concave downward on ( (-3,3) ).
(type your answers in interval notation. use a comma to separate answers as needed.)
c. the function is concave downward on. it is never concave upward.
(type your answer in interval notation. use a comma to separate answers as needed.)
find 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.
a. there is an inflection point at ( x= ).
(type an integer or a decimal. use a comma to separate answers as needed.)
b. there are no inflection points.

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.