Find the critical number(s), if any, of the function $f ( x ) = x ^ { 3 } - 21 x + 4$.
A) 4
B) $- 4$
C) $\pm \sqrt { 7 }$
D) $\pm 7$

Question

Quizplus · Accepted Answer

To find the critical numbers, we need to find the values of $x$ that make $f'(x) = 0$ or $f'(x)$ undefined.
Taking the derivative of $f(x)$, we have:

$$f'(x) = 3x^2 - 21 = 3(x^2 - 7)$$

Setting $f'(x) = 0$, we get:

$$3(x^2 - 7) = 0$$

$$x^2 - 7 = 0$$

$$x = \pm\sqrt{7}$$

So, the critical numbers are $\pm\sqrt{7}$. These are the only critical numbers since the derivative $f'(x)$ is defined and continuous for all $x$.

Therefore, the answer is $\boxed{	extbf{(C)}}$

Find the Critical Number(s), If Any, of the Function f(x)=x3−21x+4f ( x ) = x ^ { 3 } - 21 x + 4f(x)=x3−21x+4