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}-72 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}-72 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 of (f(x))

Using the power rule ((x^n)^\prime = nx^{n - 1}), for (f(x)=x^{4}-72x^{2}), we have (f^\prime(x)=4x^{3}-144x = 4x(x^{2}-36)=4x(x - 6)(x + 6))

Step2: Find the critical points

Set (f^\prime(x)=0), then (4x(x - 6)(x + 6)=0). Solving for (x), we get (x=-6), (x = 0), (x=6)

Step3: Test the intervals

  • For the interval ((-\infty,-6)), let (x=-7). Then (f^\prime(-7)=4\times(-7)\times((-7)-6)\times((-7)+6)=4\times(-7)\times(-13)\times(-1)<0)
  • For the interval ((-6,0)), let (x=-1). Then (f^\prime(-1)=4\times(-1)\times((-1)-6)\times((-1)+6)=4\times(-1)\times(-7)\times5>0)
  • For the interval ((0,6)), let (x = 1). Then (f^\prime(1)=4\times1\times(1 - 6)\times(1 + 6)=4\times1\times(-5)\times7<0)
  • For the interval ((6,\infty)), let (x = 7). Then (f^\prime(7)=4\times7\times(7 - 6)\times(7 + 6)=4\times7\times1\times13>0)

Answer:

The function (f(x)) is increasing on ((-6,0)\cup(6,\infty))