Find all critical numbers for the function $f ( x ) = x ^ { 4 } - 4 x ^ { 3 }$ .
A) $- 3$

B) $3$
C)0, 3
D)0
E)0, 4
F)2

G) $0 , - 3$
H)No critical numbers

Question

Find all critical numbers for the function $f ( x ) = x ^ { 4 } - 4 x ^ { 3 }$ .
A) $- 3$

B) $3$ 
C)0, 3
D)0
E)0, 4
F)2

G) $0 , - 3$ 
H)No critical numbers

Quizplus · Accepted Answer

Critical numbers occur where the derivative of the function equals 0 or is undefined. For $f(x) = x^4 - 4x^3$, the derivative is $f'(x) = 4x^3 - 12x^2$. Factoring out $4x^2$, we get $4x^2(x - 3)$. Setting this equal to 0 gives $x = 0$ and $x = 3$ as the points where the derivative is 0. The derivative is defined for all x, so these are the critical numbers.

Find All Critical Numbers for the Function f(x)=x4−4x3f ( x ) = x ^ { 4 } - 4 x ^ { 3 }f(x)=x4−4x3

Find All Critical Numbers for the Function $f ( x ) = x ^ { 4 } - 4 x ^ { 3 }$