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)=lnleft(x^{2}+36\right) ).\n( f(x) ) has a local minimum.\nsummarize 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\n(type your answer in interval notation. use a comma to separate answers as needed.)\na ( f(x) ) is concave upward on ( (-6,6) ) and concave downward on ( (-infty,-6),(6, infty) )\nb. ( f(x) ) is concave upward on \nc ( f(x) ) is concave downward on \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= )\n(use a comma to separate answers as needed.)\nb. there are no inflection points

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=lnleft(x^{2}+36\right) ).\n( f(x) ) has a local minimum.\nsummarize 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\n(type your answer in interval notation. use a comma to separate answers as needed.)\na ( f(x) ) is concave upward on ( (-6,6) ) and concave downward on ( (-infty,-6),(6, infty) )\nb. ( f(x) ) is concave upward on \nc ( f(x) ) is concave downward on \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= )\n(use a comma to separate answers as needed.)\nb. there are no inflection points

Answer

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)