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

Answer

Explanation:

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

The function is (f(x)=x^{4}-18x^{2}). Using the power rule ((x^{n})^\prime = nx^{n - 1}), the derivative (f^\prime(x)=4x^{3}-36x). Factor out (4x): (f^\prime(x)=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). These divide the real - line into the intervals ((-\infty,-3)), ((-3,0)), ((0,3)), and ((3,\infty)).

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 decreasing when (f^\prime(x)<0), the function (f(x)) is decreasing on the intervals ((-\infty,-3)) and ((0,3)).

Answer:

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