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 ) is concave upward on the subinterval(s) ( (-sqrt{5},sqrt{5}) ).\n(type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed)\nb. the function ( f ) is never concave upward.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) is concave downward on the subinterval(s) ( (-infty,-sqrt{5}),(sqrt{5},infty) ).\n(type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed)\nb. the function ( f ) is never concave downward.\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 = ).\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.

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 ) is concave upward on the subinterval(s) ( (-sqrt{5},sqrt{5}) ).\n(type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed)\nb. the function ( f ) is never concave upward.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) is concave downward on the subinterval(s) ( (-infty,-sqrt{5}),(sqrt{5},infty) ).\n(type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed)\nb. the function ( f ) is never concave downward.\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 = ).\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.

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 ((x^{n})^\prime=nx^{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). [ \begin{align*} 12x^{2}&=60\ x^{2}&=5\ x&=\pm\sqrt{5} \end{align*} ]

Step5: Determine concavity intervals

  • For (f^{\prime\prime}(x)>0): (-12x^{2}+60>0), (x^{2}<5), (-\sqrt{5}<x<\sqrt{5}). The function is concave upward on ((-\sqrt{5},\sqrt{5}))
  • For (f^{\prime\prime}(x)<0): (x^{2}>5), (x<-\sqrt{5}) or (x>\sqrt{5}). The function is concave downward on ((-\infty,-\sqrt{5})\cup(\sqrt{5},\infty))

Answer:

For the concavity upward: A. The function (f) is concave upward on the sub - interval(s) ((-\sqrt{5},\sqrt{5})) For the concavity downward: A. The function (f) is concave downward on the sub - interval(s) ((-\infty,-\sqrt{5}),(\sqrt{5},\infty)) For the inflection points: A. The function (f) has an inflection point at (x =-\sqrt{5},\sqrt{5})