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

1 Answer
Nov 4, 2015

Answer:

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#.