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 decreasing on ( (-infty,-5),(0.5) )\n(type your answer using interval notation. use a comma to separate answers as needed.)\nb. the function is never decreasing\nchoose the correct graph of ( f(x)=x^{4}-50 x^{2} ) with the correct horizontal tangent line below.

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 decreasing on ( (-infty,-5),(0.5) )\n(type your answer using interval notation. use a comma to separate answers as needed.)\nb. the function is never decreasing\nchoose the correct graph of ( f(x)=x^{4}-50 x^{2} ) with the correct horizontal tangent line below.

Answer

Explanation:

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

Given (f(x)=x^{4}-50x^{2}). Using the power rule ((x^{n})^\prime = nx^{n - 1}), 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). (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). So (f(x)) is decreasing on ((-\infty,-5)).
  • 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). So (f(x)) is increasing on ((-5,0)).
  • For the interval ((0,5)): Let (x = 1), then (f^\prime(1)=4\times1\times(1^{2}-25)=4\times1\times(-24)=-96<0). So (f(x)) is decreasing on ((0,5)).
  • For the interval ((5,\infty)): Let (x = 6), then (f^\prime(6)=4\times6\times(6^{2}-25)=4\times6\times11 = 264>0). So (f(x)) is increasing on ((5,\infty)).

The function (f(x)) is increasing on ((-5,0)\cup(5,\infty)) and decreasing on ((-\infty,-5)\cup(0,5)).

For the graph, the function (y = x^{4}-50x^{2}) is an even - function ((f(-x)=(-x)^{4}-50(-x)^{2}=x^{4}-50x^{2}=f(x))). The horizontal tangent lines occur at (x=-5,x = 0,x = 5).

Answer:

The function (f(x)) is increasing on ((-5,0)\cup(5,\infty)) and decreasing on ((-\infty,-5)\cup(0,5)).