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A differentiable function attains its average slope at some interior point.
Mean value theorem: Let , and let be continuous on and differentiable on . Then there exists such that
The theorem follows from Rolle's theorem by applying Rolle to a suitable affine adjustment of . It yields useful corollaries such as the mean value estimate and monotonicity criteria from the sign of the derivative.
A function is differentiable at if the finite limit
exists. It is differentiable on if it is differentiable at every point of , with one-sided limits used at endpoints of an interval.
Rolle's theorem: Let be continuous on and differentiable on . If , then there exists such that
This is a foundational consequence of the existence of a local extremum for continuous functions on compact intervals, and it is the key ingredient in the mean value theorem.
Mean value estimate lemma: Let be continuous on and differentiable on . If there exists such that for all , then