Question

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=ln left(x^{2}+36
ight) ).
find inflection points. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the inflection point(s) is(are) ( x= ) -6,6
(use a comma to separate answers as needed.)
b. there are no inflection points.
choose the correct graph below.

Explanation:

Step1: Find the first - derivative

Use the chain rule. If \(y = \ln(u)\) and \(u=x^{2}+36\), then \(y^\prime=\frac{u^\prime}{u}\).
Since \(u^\prime = 2x\), \(f^\prime(x)=\frac{2x}{x^{2}+36}\).

Step2: Find the second - derivative

Use the quotient rule \((\frac{v}{w})^\prime=\frac{v^\prime w - vw^\prime}{w^{2}}\), where \(v = 2x\), \(v^\prime=2\), \(w=x^{2}+36\), \(w^\prime = 2x\).
\(f^{\prime\prime}(x)=\frac{2(x^{2}+36)-2x(2x)}{(x^{2}+36)^{2}}=\frac{72 - 2x^{2}}{(x^{2}+36)^{2}}\).

Step3: Find the inflection points

Set \(f^{\prime\prime}(x)=0\), then \(72-2x^{2}=0\).
\(2x^{2}=72\), \(x^{2}=36\), \(x=\pm6\).

Answer:

A. The inflection point(s) is(are) \(x=-6,6\)