Question

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

Answer 1

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

`f(x) = 3x^2 - 12x +7`, the domain is all real numbers.

`f'(x)=3x^2-12`

Now find the numbers for which `f'(c)=0` or `f'(c)` does not exist.

Clearly `f'(x)` exists for all real numbers `x`.

`f'(x)=3x^2-12 = 0` where `3(x^2-4) = 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 `(c, f(c))`. In that usage, we would need to find `f(-2)` and `f(2)` to finish.)