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 maximum at ( x = -sqrt{15},sqrt{15} ).\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nb. the function f has no local maximum.\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 = 0 ).\n(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) \n(type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers \nob. the function f is never concave upward.

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 maximum at ( x = -sqrt{15},sqrt{15} ).\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nb. the function f has no local maximum.\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 = 0 ).\n(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) \n(type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers \nob. the function f is never concave upward.

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=-4x(x^{2}-15)=-4x(x-\sqrt{15})(x + \sqrt{15})) Set (f^\prime(x)=0), then (x = 0,x=\sqrt{15},x=-\sqrt{15})

Step3: Use the first - derivative test

  • For (x<-\sqrt{15}), let (x=-4), then (f^\prime(-4)=-4\times(-4)\times((-4)^{2}-15)=16\times1>0)
  • For (-\sqrt{15}<x<0), let (x=-1), then (f^\prime(-1)=-4\times(-1)\times((-1)^{2}-15)=4\times(-14)<0)
  • For (0<x<\sqrt{15}), let (x = 1), then (f^\prime(1)=-4\times1\times(1^{2}-15)=-4\times(-14)>0)
  • For (x>\sqrt{15}), let (x = 4), then (f^\prime(4)=-4\times4\times(4^{2}-15)=-16\times1<0)

Since (f^\prime(x)) changes sign from positive to negative at (x=-\sqrt{15}) and (x=\sqrt{15}), the function (f(x)) has local maxima at (x =-\sqrt{15}) and (x=\sqrt{15})

Since (f^\prime(x)) changes sign from negative to positive at (x = 0), the function (f(x)) has a local minimum at (x = 0)

Step4: Find the second - derivative

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

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

The function (f(x)) is concave upward on the interval ((-\sqrt{5},\sqrt{5}))

Answer:

For the local maximum: A. The function (f) has a local maximum at (x=-\sqrt{15},\sqrt{15}) For the local minimum: A. The function (f) has a local minimum at (x = 0) For the concavity: A. The function (f) is concave upward on the sub - interval((-\sqrt{5},\sqrt{5}))