find the intervals on which ( f(x) ) is increasing and the intervals on which ( f(x) ) is decreasing. then…

find the intervals on which ( f(x) ) is increasing and the intervals on which ( f(x) ) is decreasing. then sketch the graph. add horizontal tangent lines.\n( f(x)=x^{4}-18 x^{2} )\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function is increasing on\n(type your answer using interval notation. use a comma to separate answers as needed.)\nb. the function is never increasing.

find the intervals on which ( f(x) ) is increasing and the intervals on which ( f(x) ) is decreasing. then sketch the graph. add horizontal tangent lines.\n( f(x)=x^{4}-18 x^{2} )\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function is increasing on\n(type your answer using interval notation. use a comma to separate answers as needed.)\nb. the function is never increasing.

Answer

Explanation:

Step1: Find the derivative

Given ( f(x)=x^{4}-18x^{2} ), using the power rule ((x^n)^\prime = nx^{n - 1}), we have ( f^\prime(x)=4x^{3}-36x=4x(x^{2}-9)=4x(x - 3)(x + 3) ).

Step2: Find the critical points

Set ( f^\prime(x)=0 ), then ( 4x(x - 3)(x + 3)=0 ). Solving for ( x ), we get ( x=-3,0,3 ).

Step3: Test the intervals

  • For ( x<-3 ), let ( x=-4 ), then ( f^\prime(-4)=4\times(-4)\times((-4)-3)\times((-4)+3)=4\times(-4)\times(-7)\times(-3)=-336<0 ).
  • For ( -3<x<0 ), let ( x=-1 ), then ( f^\prime(-1)=4\times(-1)\times((-1)-3)\times((-1)+3)=4\times(-1)\times(-4)\times2 = 32>0 ).
  • For ( 0<x<3 ), let ( x = 1 ), then ( f^\prime(1)=4\times1\times(1 - 3)\times(1 + 3)=4\times1\times(-2)\times4=-32<0 ).
  • For ( x>3 ), let ( x = 4 ), then ( f^\prime(4)=4\times4\times(4 - 3)\times(4 + 3)=4\times4\times1\times7 = 112>0 ).

Answer:

The function ( f(x) ) is increasing on ( (-3,0)\cup(3,\infty) ).