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}-50 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}-50 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}-50x^{2}), we have (f^\prime(x)=4x^{3}-100x = 4x(x^{2}-25)=4x(x - 5)(x + 5))

Step2: Find the critical points

Set (f^\prime(x)=0). Then (4x(x - 5)(x + 5)=0). Solving for (x), we get (x=- 5,x = 0,x = 5)

Step3: Test the intervals

  • For the interval ((-\infty,-5)), let (x=-6). Then (f^\prime(-6)=4\times(-6)\times((-6)^{2}-25)=4\times(-6)\times11=-264<0)
  • For the interval ((-5,0)), let (x=-1). Then (f^\prime(-1)=4\times(-1)\times((-1)^{2}-25)=4\times(-1)\times(-24) = 96>0)
  • For the interval ((0,5)), let (x = 1). Then (f^\prime(1)=4\times1\times(1^{2}-25)=4\times1\times(-24)=-96<0)
  • For the interval ((5,\infty)), let (x = 6). Then (f^\prime(6)=4\times6\times(6^{2}-25)=4\times6\times11 = 264>0)

Answer:

A. The function is increasing on ((-5,0),(5,\infty))