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.

[edit] 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-r<y<x we have f'(y)>0, and for every y such that x<y<x+r we have f'(y)<0, then f has a maximum 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-r<y<x we have f'(y)<0, and for every y such that x<y<x+r we have f'(y)>0, 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-r<y<x or x<y<x+r we have either f'(y)>0 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.

[edit] See also

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