Question

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=ln left(x^{2}+36
ight) ).
(type your answer in interval notation. use a comma to separate answers as needed.)
a. ( f(x) ) is increasing on ( (0, infty) ) and decreasing on ( (-infty, 0) ).
b. ( f(x) ) is increasing on
c. ( f(x) ) is decreasing on
( 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 and concave downward on
b. ( f(x) ) is concave upward on
c. ( f(x) ) is concave downward on

Explanation:

Step1: Find the first - derivative

We know that if \(y = \ln(u)\), then \(y^\prime=\frac{u^\prime}{u}\). Let \(u=x^{2}+36\), then \(u^\prime = 2x\). So \(f^\prime(x)=\frac{2x}{x^{2}+36}\).
Set \(f^\prime(x)=0\), we get \(2x = 0\), so \(x = 0\).
When \(x>0\), \(f^\prime(x)=\frac{2x}{x^{2}+36}>0\); when \(x < 0\), \(f^\prime(x)=\frac{2x}{x^{2}+36}<0\).

Step2: Find the second - derivative

Using the quotient rule \((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}\), where \(u = 2x\), \(u^\prime=2\), \(v=x^{2}+36\), \(v^\prime = 2x\).
\(f^{\prime\prime}(x)=\frac{2(x^{2}+36)-2x\times(2x)}{(x^{2}+36)^{2}}=\frac{72 - 2x^{2}}{(x^{2}+36)^{2}}\).
Set \(f^{\prime\prime}(x)=0\), then \(72-2x^{2}=0\), \(x^{2}=36\), \(x=\pm6\).
When \(x\in(-6,6)\), \(f^{\prime\prime}(x)=\frac{72 - 2x^{2}}{(x^{2}+36)^{2}}>0\); when \(x\in(-\infty,-6)\cup(6,\infty)\), \(f^{\prime\prime}(x)=\frac{72 - 2x^{2}}{(x^{2}+36)^{2}}<0\).

Answer:

A. \(f(x)\) is concave upward on \((-6,6)\) and concave downward on \((-\infty,-6),(6,\infty)\)