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}-72 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 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 \((-6,0)\cup(6,\infty)\)