# How do you find all the critical points of the function f(x) = x^3 − 12x + 7?

Mar 30, 2015

A critical point for function $f$ is a number $c$ that
(1) is in the domain of $f$ and
(2) has $f ' \left(c\right) = 0$ or $f ' \left(c\right)$ does not exist.

$f \left(x\right) = 3 {x}^{2} - 12 x + 7$, the domain is all real numbers.

$f ' \left(x\right) = 3 {x}^{2} - 12$

Now find the numbers for which $f ' \left(c\right) = 0$ or $f ' \left(c\right)$ does not exist.

Clearly $f ' \left(x\right)$ exists for all real numbers $x$.

$f ' \left(x\right) = 3 {x}^{2} - 12 = 0$ where $3 \left({x}^{2} - 4\right) = 0$ which happens where ${x}^{2} = 4$ and that's at $- 2$ and at $2$

Because $- 2$ and $2$ are both also in the domain of $f$, they are both critical numbers (points) for $f$.

(I believe that some people use "critical point to mean a point $\left(c , f \left(c\right)\right)$. In that usage, we would need to find $f \left(- 2\right)$ and $f \left(2\right)$ to finish.)