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 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 \(f(x)=x^{4}-50x^{2}\), 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 \((-5,0),(5,\infty)\)