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 $f(x)=\\frac{3x^{2}+1}{x^{2}-9}$.\n(type an equation.)\nb. the function has two vertical asymptotes. the leftmost asymptote is $x = - 3$ and the rightmost asymptote is $x = 3$.\n(type equations.)\nc. there are no vertical asymptotes.\nfind the intervals where $f(x)$ is increasing or decreasing. select the correct choice below and fill in the answer box(es) to complete your choice.\na. the function is increasing on. it is never decreasing.\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function is increasing on. it is decreasing on.\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.)

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $f(x)=\\frac{3x^{2}+1}{x^{2}-9}$.\n(type an equation.)\nb. the function has two vertical asymptotes. the leftmost asymptote is $x = - 3$ and the rightmost asymptote is $x = 3$.\n(type equations.)\nc. there are no vertical asymptotes.\nfind the intervals where $f(x)$ is increasing or decreasing. select the correct choice below and fill in the answer box(es) to complete your choice.\na. the function is increasing on. it is never decreasing.\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function is increasing on. it is decreasing on.\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.)

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}}), where (u = 3x^{2}+1), (u^\prime=6x), (v=x^{2}-9), (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), (\frac{-56x}{(x^{2}-9)^{2}} = 0), which gives (x = 0) (since the denominator ((x^{2}-9)^{2}\neq0) for (x\neq\pm3)).

Step3: Determine the sign of (f^\prime(x)) in different intervals

  • For (x\lt0) (say (x=-1)), (f^\prime(-1)=\frac{-56\times(-1)}{((-1)^{2}-9)^{2}}=\frac{56}{64}\gt0)
  • For (x\gt0) (say (x = 1)), (f^\prime(1)=\frac{-56\times1}{(1^{2}-9)^{2}}=\frac{-56}{64}\lt0)

Answer:

B. The function is increasing on ((-\infty,- 3),(-3,0)). It is decreasing on ((0,3),(3,\infty))