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\n( f(x)=(x^{2}+6)(36 - x^{2}) )\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the function ( f ) is increasing on the subinterval(s) ( (-infty,-sqrt{15}),(0,sqrt{15}) ).\n(type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\n\nb. the function ( f ) is never increasing.\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the function ( f ) is decreasing on the subinterval(s) ( (-sqrt{15},0),(sqrt{15},infty) ).\n(type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\n\nb. the function ( f ) is never decreasing.\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the function ( f ) has a local maximum at ( x = ).\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\n\nb. the function ( f ) has no local maximum.

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n\n( f(x)=(x^{2}+6)(36 - x^{2}) )\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the function ( f ) is increasing on the subinterval(s) ( (-infty,-sqrt{15}),(0,sqrt{15}) ).\n(type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\n\nb. the function ( f ) is never increasing.\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the function ( f ) is decreasing on the subinterval(s) ( (-sqrt{15},0),(sqrt{15},infty) ).\n(type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\n\nb. the function ( f ) is never decreasing.\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the function ( f ) has a local maximum at ( x = ).\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\n\nb. the function ( f ) has no local maximum.

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), we have (f^\prime(x)=-4x^{3}+60x=-4x(x^{2}-15)=-4x(x-\sqrt{15})(x + \sqrt{15}))

Step3: Find the critical points

Set (f^\prime(x)=0), then (-4x(x-\sqrt{15})(x + \sqrt{15})=0). The critical points are (x = 0), (x=\sqrt{15}), and (x=-\sqrt{15})

Step4: Determine the intervals of increase and decrease

  • Test the interval ((-\infty,-\sqrt{15})): Let (x=-4), then (f^\prime(-4)=-4\times(-4)\times((-4)^{2}-15)=16\times1>0). So (f(x)) is increasing on ((-\infty,-\sqrt{15}))
  • Test the interval ((-\sqrt{15},0)): Let (x=-1), then (f^\prime(-1)=-4\times(-1)\times((-1)^{2}-15)=4\times(-14)<0). So (f(x)) is decreasing on ((-\sqrt{15},0))
  • Test the interval ((0,\sqrt{15})): Let (x = 1), then (f^\prime(1)=-4\times1\times(1^{2}-15)=-4\times(-14)>0). So (f(x)) is increasing on ((0,\sqrt{15}))
  • Test the interval ((\sqrt{15},\infty)): Let (x = 4), then (f^\prime(4)=-4\times4\times(4^{2}-15)=-16\times1<0). So (f(x)) is decreasing on ((\sqrt{15},\infty))

Step5: Find local maxima

Since the function changes from increasing to decreasing at (x=-\sqrt{15}) and (x = \sqrt{15}) (f(-\sqrt{15})=-(-\sqrt{15})^{4}+30(-\sqrt{15})^{2}+216=-225 + 450+216=441) (f(\sqrt{15})=-(\sqrt{15})^{4}+30(\sqrt{15})^{2}+216=-225 + 450+216=441)

Answer:

  • For the increasing intervals: (A). The function (f) is increasing on the sub - interval(s) ((-\infty,-\sqrt{15}),(0,\sqrt{15}))
  • For the decreasing intervals: (A). The function (f) is decreasing on the sub - interval(s) ((-\sqrt{15},0),(\sqrt{15},\infty))
  • For local maxima: (A). The function (f) has a local maximum at (x=-\sqrt{15},\sqrt{15})