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)=ln left(x^{2}+36\right) ).\nfind inflection points. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the inflection point(s) is(are) ( x= ) -6,6\n(use a comma to separate answers as needed.)\nb. there are no inflection points.\nchoose the correct graph below.\n

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

Answer

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)