Question

determine all critical points for the function.
f(x)=3x^{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 \(y = f(x)=3x^{2}-2x^{3}\), the derivative \(f^\prime(x)\) is:

$$
LATEXBLOCK0
$$

Step2: Set the derivative equal to zero

Set \(f^\prime(x)=0\), so \(6x(1 - x)=0\).
By the zero - product property \(ab = 0\) implies \(a = 0\) or \(b = 0\).
If \(6x=0\), then \(x = 0\); if \(1 - x=0\), then \(x = 1\).

Answer:

\(x = 0,1\)