Question

determine all critical points for the function.

f(x)=-12x^{2}+2x^{3}

x=
(use a comma to separate answers as needed.)

Explanation:

Step1: Find the derivative of the function

Using the power rule \((x^n)^\prime=nx^{n - 1}\), for \(y = f(x)=-12x^{2}+2x^{3}\), the derivative \(f^\prime(x)=(-12x^{2})^\prime+(2x^{3})^\prime\).
\(f^\prime(x)=-12\times2x+2\times3x^{2}=-24x + 6x^{2}\).

Step2: Set the derivative equal to zero

Set \(f^\prime(x)=0\), so \(6x^{2}-24x=0\). Factor out the common factor \(6x\): \(6x(x - 4)=0\).

Step3: Solve for \(x\)

Using the zero - product property \(ab = 0\) implies \(a = 0\) or \(b = 0\).
If \(6x=0\), then \(x = 0\); if \(x - 4=0\), then \(x=4\).

Answer:

\(0,4\)