Determine all critical points for the function.
-$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 9$
A) $x = 0$ and $x = 2$
B) $x = - 2$ and $x = 2$
C) $x = 0$
D) $x = 0$ and $x = 4$

Question

Quizplus · Accepted Answer

To find the critical points, we need to find where the derivative equals 0 or does not exist.

$$f'(x) = 3x^2 - 12x$$
Setting the derivative equal to 0:
$$3x^2 - 12x = 0$$
$$3x(x-4) = 0$$
So, the critical points are when $x=0$ and $x=4$. Thus, the correct choice is D.

Determine All Critical Points for the Function f(x)=x3−6x2+9f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 9f(x)=x3−6x2+9

Determine All Critical Points for the Function $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 9$