Question

What is the derivative of a function equal to at a critical point?

Answer 1

It is either 0 or it does not exist.

Explanation:

A critical number for a function is a number in the domain of the function at which the derivative is either 0 or fails to exist.

Examples

`f(x) = x^3+5x^2-7` has derivative `f'(x) = 3x^2+10x` and critical numbers `0` and `-10/3`

`g(x) = x/(x-5)` has derivative `g'(x)=(-5)/(x-5)^2` which is never `0` and is undefined only at `5` which is not in the domain of `g`. So, `g` has no critical numbers.

`h(x) = (x-1)^(2/3)` has derivative `h'(x)=2/(3(x-1)^(1/3)) = 2/(3root(3)(x-1))` which is never `0` and is undefined only at `1` which is in the domain of `h`. So, `h` has critical number `1`.