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 ( x )-intercept(s) of ( f ) is/are ( x = -6,6 ).\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nb. the function ( f ) has no ( x )-intercepts.\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) \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 increasing.

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 ( x )-intercept(s) of ( f ) is/are ( x = -6,6 ).\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nb. the function ( f ) has no ( x )-intercepts.\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) \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 increasing.

Answer

Explanation:

Step1: Find the x - intercepts

Set (y = f(x)=0), so ((x^{2}+6)(36 - x^{2})=0). Since (x^{2}+6>0) for all real (x) (because (x^{2}\geq0), then (x^{2}+6\geq6)), we solve (36 - x^{2}=0). Using the difference of squares formula (a^{2}-b^{2}=(a + b)(a - b)), where (a = 6) and (b=x), we have ((6 + x)(6 - x)=0). Set each factor equal to zero: (6+x=0) gives (x=-6), and (6 - x=0) gives (x = 6).

Step2: Find the y - intercepts

Set (x = 0) in (y=f(x)). Then (y=(0^{2}+6)(36-0^{2})=6\times36 = 216).

Step3: Analyze the function's increasing/decreasing behavior

First, expand (f(x)=(x^{2}+6)(36 - x^{2})=36x^{2}-x^{4}+216 - 6x^{2}=-x^{4}+30x^{2}+216). Take the derivative (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 a sign - chart for (f^\prime(x)):

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

So the function (f(x)) is increasing on the intervals ((-\infty,-\sqrt{15})) and ((0,\sqrt{15})).

Answer:

For the x - intercepts: A. The x - intercept(s) of (f) is/are (x=-6,6). For the y - intercepts: A. The y - intercept of (f) is (y = 216). For the increasing intervals: A. The function (f) is increasing on the subinterval(s) ((-\infty,-\sqrt{15}),(0,\sqrt{15})).