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 a local minimum at ( x = ) (type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nb. the function ( f ) has no local minimum.\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) (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)\nb. the function ( f ) is never concave downward.

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 a local minimum at ( x = ) (type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nb. the function ( f ) has no local minimum.\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) (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)\nb. the function ( f ) is never concave downward.

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}), for (y=-x^{4}+30x^{2}+216) (f^\prime(x)=-4x^{3}+60x=-4x(x^{2}-15)=-4x(x-\sqrt{15})(x + \sqrt{15})) Set (f^\prime(x)=0), we get (x = 0,x=\sqrt{15},x=-\sqrt{15}) Use the first - derivative test:

  • For (x<-\sqrt{15}), let (x=-4), (f^\prime(-4)=-4\times(-4)\times((-4)^{2}-15)=16\times1>0)
  • For (-\sqrt{15}<x<0), let (x = - 1), (f^\prime(-1)=-4\times(-1)\times((-1)^{2}-15)=4\times(-14)<0)
  • For (0<x<\sqrt{15}), let (x = 1), (f^\prime(1)=-4\times1\times(1^{2}-15)=-4\times(-14)>0)
  • For (x>\sqrt{15}), let (x = 4), (f^\prime(4)=-4\times4\times(4^{2}-15)=-16\times1<0) Since the function changes from decreasing ((-\sqrt{15}<x<0)) to increasing ((0<x<\sqrt{15})), (x = 0) is a local minimum.

Step3: Find the second - derivative

Differentiate (f^\prime(x)=-4x^{3}+60x) (f^{\prime\prime}(x)=-12x^{2}+60=-12(x^{2}-5)=-12(x-\sqrt{5})(x + \sqrt{5})) Set (f^{\prime\prime}(x)=0), we get (x=\sqrt{5},x = -\sqrt{5})

  • For (x<-\sqrt{5}), let (x=-3), (f^{\prime\prime}(-3)=-12\times((-3)^{2}-5)=-12\times4<0)
  • For (-\sqrt{5}<x<\sqrt{5}), let (x = 0), (f^{\prime\prime}(0)=-12\times(0 - 5)=60>0)
  • For (x>\sqrt{5}), let (x = 3), (f^{\prime\prime}(3)=-12\times(9 - 5)=-48<0)

Answer:

For the local - minimum question: A. The function (f) has a local minimum at (x = 0) For the concave - upward question: A. The function (f) is concave upward on the subinterval(s) ((-\sqrt{5},\sqrt{5})) For the concave - downward question: A. The function (f) is concave downward on the subinterval(s) ((-\infty,-\sqrt{5})\cup(\sqrt{5},\infty))