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}-18 x^{2} )
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function is increasing on
(type your answer using interval notation. use a comma to separate answers as needed.)
b. the function is never increasing.

Explanation:

Step1: Find the derivative

Given \( f(x)=x^{4}-18x^{2} \), using the power rule \((x^n)^\prime = nx^{n - 1}\), we have \( f^\prime(x)=4x^{3}-36x=4x(x^{2}-9)=4x(x - 3)(x + 3) \).

Step2: Find the critical points

Set \( f^\prime(x)=0 \), then \( 4x(x - 3)(x + 3)=0 \). Solving for \( x \), we get \( x=-3,0,3 \).

Step3: Test the intervals

• For \( x<-3 \), let \( x=-4 \), then \( f^\prime(-4)=4\times(-4)\times((-4)-3)\times((-4)+3)=4\times(-4)\times(-7)\times(-3)=-336<0 \).
• For \( -3
• For \( 0
• For \( x>3 \), let \( x = 4 \), then \( f^\prime(4)=4\times4\times(4 - 3)\times(4 + 3)=4\times4\times1\times7 = 112>0 \).

Answer:

The function \( f(x) \) is increasing on \( (-3,0)\cup(3,\infty) \).