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 ( y )-intercept of ( f ) is ( y = 216 ).\n(type an exact answer, using radicals as needed.)\nb. the function ( f ) has no ( y )-intercept.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\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 nee\nb. the function ( f ) is never increasing.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) is decreasing 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 as needed\nb. the function ( f ) is never decreasing.

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 ( y )-intercept of ( f ) is ( y = 216 ).\n(type an exact answer, using radicals as needed.)\nb. the function ( f ) has no ( y )-intercept.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\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 nee\nb. the function ( f ) is never increasing.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) is decreasing 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 as needed\nb. the function ( f ) is never decreasing.

Answer

Explanation:

Step 1: Expand the function

First, expand (f(x)=(x^{2}+6)(36 - x^{2})): [ \begin{align*} f(x)&=x^{2}\times36-x^{2}\times x^{2}+6\times36 - 6\times x^{2}\ &=36x^{2}-x^{4}+216-6x^{2}\ &=-x^{4}+30x^{2}+216 \end{align*} ]

Step 2: Find the derivative

Differentiate (y = f(x)=-x^{4}+30x^{2}+216) using the power rule ((x^{n})^\prime=nx^{n - 1}). The derivative (f^\prime(x)=-4x^{3}+60x=-4x(x^{2}-15)=-4x(x-\sqrt{15})(x + \sqrt{15}))

Step 3: Determine the intervals of increase and decrease

Set (f^\prime(x)>0) to find where the function is increasing: (-4x(x-\sqrt{15})(x+\sqrt{15})>0) The critical points are (x =-\sqrt{15},0,\sqrt{15}) Using a sign - chart:

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

Set (f^\prime(x)<0) to find where the function is decreasing. The function (f(x)) is decreasing on the intervals ((-\sqrt{15},0)) and ((\sqrt{15},\infty))

Answer:

The function (f) is decreasing on the sub - intervals (\left(-\sqrt{15},0\right),\left(\sqrt{15},\infty\right))