Question

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.
( f(x)=x^{4}-50 x^{2} )
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function is decreasing on ( (-infty,-5),(0.5) )
(type your answer using interval notation. use a comma to separate answers as needed.)
b. the function is never decreasing
choose the correct graph of ( f(x)=x^{4}-50 x^{2} ) with the correct horizontal tangent line below.

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 \((-\infty,-5)\cup(0,5)\).