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 ( (-3,0),(3, infty) ).\n(type your answer using interval notation. use a comma to separate answers as needed.)\nb. the function is never increasing.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function is decreasing on\n(type your answer using interval notation. use a comma to separate answers as needed.)\nb. the function is never decreasing.

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 ( (-3,0),(3, infty) ).\n(type your answer using interval notation. use a comma to separate answers as needed.)\nb. the function is never increasing.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function is decreasing on\n(type your answer using interval notation. use a comma to separate answers as needed.)\nb. the function is never decreasing.

Answer

Explanation:

Step1: Find the derivative of (f(x))

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), (x = 0), and (x = 3).

Step3: Test the intervals

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

Since the function is increasing when (f^\prime(x)>0) (intervals ((-3,0)) and ((3,\infty))) and decreasing when (f^\prime(x)<0). The intervals where (f(x)) is decreasing are ((-\infty,-3)) and ((0,3)).

Answer:

The function is decreasing on ((-\infty,-3),(0,3))