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

1 Answer
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'(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.)