$\text { Find all critical numbers of the function } g ( x ) = x ^ { 4 } - 4 x ^ { 2 } \text {. }$
A) critical numbers: $x = 0 , x = 2 \sqrt { 2 } , x = - 2 \sqrt { 2 }$
B) critical numbers: $x = 0 , x = \sqrt { 2 } , x = - \sqrt { 2 }$
C) critical numbers: $x = 2 \sqrt { 2 } , x = - 2 \sqrt { 2 }$
D) critical numbers: $x = \sqrt { 2 } , x = - \sqrt { 2 }$
E) no critical numbers

Question

$\text { Find all critical numbers of the function } g ( x ) = x ^ { 4 } - 4 x ^ { 2 } \text {. }$ 
A) critical numbers: $x = 0 , x = 2 \sqrt { 2 } , x = - 2 \sqrt { 2 }$
B) critical numbers: $x = 0 , x = \sqrt { 2 } , x = - \sqrt { 2 }$
C) critical numbers: $x = 2 \sqrt { 2 } , x = - 2 \sqrt { 2 }$
D) critical numbers: $x = \sqrt { 2 } , x = - \sqrt { 2 }$
E) no critical numbers

Quizplus · Accepted Answer

We find the critical numbers by finding the values of $x$ where $g'(x) = 0$ or $g'(x)$ does not exist.
$$g(x) = x^4 - 4x^2$$
$$g'(x) = 4x^3 - 8x$$
Setting $g'(x) = 0$, we get
$$4x(x^2 - 2) = 0$$
This gives us critical numbers at $x = 0$, $x = \sqrt{2}$, and $x = -\sqrt{2}$. 
Note that $g'(x)$ exists for all $x$, so these are the only critical numbers. Thus, the correct choice is $\boxed{	extbf{(B)}}$.

Find all critical numbers of the function g(x)=x4−4x2. \text { Find all critical numbers of the function } g ( x ) = x ^ { 4 } - 4 x ^ { 2 } \text {. } Find all critical numbers of the function g(x)=x4−4x2.

$\text { Find all critical numbers of the function } g ( x ) = x ^ { 4 } - 4 x ^ { 2 } \text {. }$