First derivative test

Calculus

A method for determining whether a critical point is a minimum, maximum, or neither. 1. If f'(x) > 0 on an open interval extending left from x₀ and f'(x) < 0 on an open interval extending right from x₀, then f(x) has a local maximum (possibly a global maximum) at x₀.

2. If f'(x) < 0 on an open interval extending left from x₀ and f'(x) > 0 on an open interval extending right from x₀, then f(x) has a local minimum (possibly a global minimum) at x₀.

3. If f'(x) has the same sign on an open interval extending left from x₀ and on an open interval extending right from x₀, then f(x) has an inflection point at x₀.

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