First derivative test

In calculus, a branch of mathematics, the first derivative test determines whether a given critical point of a function is a maximum, a minimum, or neither.

Introduction

Suppose that "f" is a function and we want to determine if "f" has a maximum or minimum at "x". If "x" is a maximum of "f", then "f" is increasing to the left of "x" and decreasing to the right of "x". Similarly, if "x" is a minimum of "f", then "f" is decreasing to the left of "x" and increasing to the right of "x". If "f" is increasing on both sides of "x", or if "f" is decreasing on both sides of "x", then "x" is not a maximum or a minimum.

If "f" is differentiable in a neighbourhood of "x", we can rephrase the conditions of being increasing or decreasing in terms of the derivative of "f". When the derivative of "f" is positive, then "f" is increasing, and when the derivative of "f" is negative, then "f" is decreasing. The first derivative test now states:

*If there exists a positive number "r" such that "f' " is continuous between "x-r" and "x+r", and for every "y" such that "x-r0", and for every "y" such that "x*If there exists a positive number "r" such that "f' " is continuous between "x-r" and "x+r", and for every "y" such that "x-r0", then "f" has a minimum at "x".
*If there exists a positive number "r" such that "f' " is continuous between "x-r" and "x+r", and for every "y" such that "x-r0" or "f'(y)<0", then "f" has neither a maximum nor a minimum at "x".
*If "f' " is not continuous between "x-r" and "x+r" for any "r", or if none of the above conditions hold for any "r" for which "f' " is continuous between "x-r" and "x+r", then the test fails.

ee also

* Fermat's theorem
* Second derivative test
* Higher order derivative test