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

1 Answer
Jim H
Nov 4, 2015

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.

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