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} ).
(type your answers in interval notation. use a comma to separate answers as needed.)
c. the function is decreasing on . it is never increasing.
(type your answer in interval notation. use a comma to separate answers as needed.)
find 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.
a. there is a local minimum at ( x= ). there is no local maximum.
(type an integer or a decimal. use a comma to separate answers as needed.)
b. there is a local maximum at ( x= ). there is no local minimum.
(type an integer or a decimal. use a comma to separate answers as needed.)
c. there is a local maximum at ( x= ) and there is a local minimum at ( x= ).
(type integers or decimals. use a comma to separate answers as needed.)
d. there are no local extrema.

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 \(v^\prime = 2x\).

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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 \(x
eq\pm3\), 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.