Question

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=lnleft(x^{2}+36
ight) ).
( f(x) ) has a local minimum.
summarize the pertinent information obtained by analyzing ( f^{prime prime}(x) ). select the correct choice below and fill in the answer box(es) to complete your choice
(type your answer in interval notation. use a comma to separate answers as needed.)
a ( f(x) ) is concave upward on ( (-6,6) ) and concave downward on ( (-infty,-6),(6, infty) )
b. ( f(x) ) is concave upward on
c ( f(x) ) is concave downward on
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= )
(use a comma to separate answers as needed.)
b. there are no inflection points

Explanation:

Step1: Find the first - derivative

Using 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

Using 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 concavity

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

• For \(x\in(-\infty,-6)\), let \(x=-7\), \(f^{\prime\prime}(-7)=\frac{72-2\times49}{(49 + 36)^{2}}=\frac{72 - 98}{85^{2}}<0\).
• For \(x\in(-6,6)\), let \(x = 0\), \(f^{\prime\prime}(0)=\frac{72-0}{36^{2}}>0\).
• For \(x\in(6,\infty)\), let \(x = 7\), \(f^{\prime\prime}(7)=\frac{72-2\times49}{(49 + 36)^{2}}=\frac{72 - 98}{85^{2}}<0\).

Step4: Find inflection points

Since \(f^{\prime\prime}(x)\) changes sign at \(x=-6\) and \(x = 6\).
When \(x=-6\), \(y=\ln(36 + 36)=\ln(72)\).
When \(x = 6\), \(y=\ln(36 + 36)=\ln(72)\).

Answer:

A. \(f(x)\) is concave upward on \((-6,6)\) and concave downward on \((-\infty,-6),(6,\infty)\)
A. The inflection point(s) is(are) \(x=-6,6\)