determine all critical points for the function.\nf(x)=3x^{2}-2x^{3}\nx = \n(use a comma to separate answers…

determine all critical points for the function.\nf(x)=3x^{2}-2x^{3}\nx = \n(use a comma to separate answers as needed.)

determine all critical points for the function.\nf(x)=3x^{2}-2x^{3}\nx = \n(use a comma to separate answers as needed.)

Answer

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: [ \begin{align*} f^\prime(x)&=(3x^{2})^\prime-(2x^{3})^\prime\ &=3\times2x-2\times3x^{2}\ &=6x - 6x^{2}\ &=6x(1 - x) \end{align*} ]

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)