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)=left(x^{2}+6\right)left(36 - x^{2}\right) )\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) ( (-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)\nb. the function ( f ) is never decreasing.\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 = ).\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.

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n( f(x)=left(x^{2}+6\right)left(36 - x^{2}\right) )\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) ( (-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)\nb. the function ( f ) is never decreasing.\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 = ).\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.

Answer

Explanation:

Step1: Expand the function

First, expand (f(x)=(x^{2}+6)(36 - x^{2})). Using the FOIL method: [ \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*} ]

Step2: Find the first - derivative

Differentiate (y = f(x)=-x^{4}+30x^{2}+216) with respect to (x). Using the power rule ((x^{n})^\prime=nx^{n - 1}), we have (y^\prime=f^\prime(x)=-4x^{3}+60x=-4x(x^{2}-15)=-4x(x-\sqrt{15})(x + \sqrt{15}))

Step3: Determine 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},x=-\sqrt{15})

Step4: Use the first - derivative test

  • Interval ((-\infty,-\sqrt{15})): Let (x=-4). Then (f^\prime(-4)=-4\times(-4)\times((-4)^{2}-15)=16\times1 = 16>0), so (f(x)) is increasing on ((-\infty,-\sqrt{15}))
  • Interval ((-\sqrt{15},0)): Let (x=-1). Then (f^\prime(-1)=-4\times(-1)\times((-1)^{2}-15)=4\times(-14)=-56<0), so (f(x)) is decreasing on ((-\sqrt{15},0))
  • Interval ((0,\sqrt{15})): Let (x = 1). Then (f^\prime(1)=-4\times1\times(1^{2}-15)=-4\times(-14)=56>0), so (f(x)) is increasing on ((0,\sqrt{15}))
  • Interval ((\sqrt{15},\infty)): Let (x = 4). Then (f^\prime(4)=-4\times4\times(4^{2}-15)=-16\times1=-16<0), so (f(x)) is decreasing on ((\sqrt{15},\infty))

Since the function changes from decreasing to increasing at (x = 0), by the first - derivative test, the function (f(x)) has a local minimum at (x = 0)

Answer:

For the local minimum: A. The function (f) has a local minimum at (x = 0)