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 your answers in interval notation. use a comma to separate answers as needed.)\nc. the function is decreasing on . it is never increasing.\n(type your answer in interval notation. use a comma to separate answers as needed.)\nfind 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.\na. there is a local minimum at ( x= ). there is no local maximum.\n(type an integer or a decimal. use a comma to separate answers as needed.)\nb. there is a local maximum at ( x= ). there is no local minimum.\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.

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 your answers in interval notation. use a comma to separate answers as needed.)\nc. the function is decreasing on . it is never increasing.\n(type your answer in interval notation. use a comma to separate answers as needed.)\nfind 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.\na. there is a local minimum at ( x= ). there is no local maximum.\n(type an integer or a decimal. use a comma to separate answers as needed.)\nb. there is a local maximum at ( x= ). there is no local minimum.\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.

Answer

Explanation:

Step1: Find the derivative of (f(x))

Use the quotient rule ((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}). Let (u = 3x^{2}+1), then (u^\prime=6x); let (v=x^{2}-9), then (v^\prime = 2x). [ \begin{align*} f^\prime(x)&=\frac{6x(x^{2}-9)-(3x^{2}+1)\times2x}{(x^{2}-9)^{2}}\ &=\frac{6x^{3}-54x-(6x^{3}+2x)}{(x^{2}-9)^{2}}\ &=\frac{6x^{3}-54x - 6x^{3}-2x}{(x^{2}-9)^{2}}\ &=\frac{- 56x}{(x^{2}-9)^{2}} \end{align*} ]

Step2: Find the critical points

Set (f^\prime(x)=0), then (\frac{-56x}{(x^{2}-9)^{2}} = 0). Since ((x^{2}-9)^{2}>0) for (x\neq\pm3), we have (x = 0).

Step3: Determine the intervals of increase and decrease

  • For (x<0) (e.g., (x=-1)), (f^\prime(-1)=\frac{-56\times(-1)}{((-1)^{2}-9)^{2}}=\frac{56}{64}>0)
  • For (x>0) (e.g., (x = 1)), (f^\prime(1)=\frac{-56\times1}{(1^{2}-9)^{2}}=\frac{-56}{64}<0)

The function (f(x)) is increasing on ((-\infty,0)) and decreasing on ((0,\infty))

Step4: Find local extrema

Since the function changes from increasing to decreasing at (x = 0), by the first - derivative test, there is a local maximum at (x = 0)

Answer:

B. There is a local maximum at (x = 0). There is no local minimum.