summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y =…

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n( f(x)=(x^{2}+6)(36 - x^{2}) )\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) has an inflection point at ( x = -sqrt{5},sqrt{5} )\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nb. the function ( f ) has no inflection point.\nchoose the correct graph of ( y = f(x) ) below.

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n( f(x)=(x^{2}+6)(36 - x^{2}) )\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) has an inflection point at ( x = -sqrt{5},sqrt{5} )\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nb. the function ( f ) has no inflection point.\nchoose the correct graph of ( y = f(x) ) below.

Answer

Explanation:

Step1: Expand the function

$$ \begin{align*} f(x)&=(x^{2}+6)(36 - x^{2})\ &=36x^{2}-x^{4}+216 - 6x^{2}\ &=-x^{4}+30x^{2}+216 \end{align*} $$

Step2: Find the first - derivative

Using the power rule (y = ax^{n}), (y^\prime=anx^{n - 1}), we have (f^\prime(x)=-4x^{3}+60x)

Step3: Find the second - derivative

Differentiate (f^\prime(x)) again. (f^{\prime\prime}(x)=-12x^{2}+60)

Step4: Find the inflection points

Set (f^{\prime\prime}(x) = 0), then (-12x^{2}+60 = 0) (12x^{2}=60), (x^{2}=5), (x=\pm\sqrt{5})

Answer:

A. The function (f) has an inflection point at (x =-\sqrt{5},\sqrt{5})