Mean value theorem

For a function continuous on a closed interval and differentiable inside it, some interior point has a derivative equal to the function’s average rate of change across the interval.

Rolle's theorem: A theorem stating that a differentiable function with equal endpoint values has a zero derivative at some interior point. It is the zero-average-change special case that underlies the mean value theorem.
Taylor's theorem: A theorem expressing a function near a point as a polynomial plus a controlled remainder. Its remainder estimates are proved using repeated applications of the mean value theorem.
Continuity: A property of a function whose values change without jumps at a point or throughout a domain. Continuity on the closed interval is an essential hypothesis of the theorem.
Mean value theorem for integrals: A theorem equating the average value of a continuous function on an interval with one of its values there. It matches an integral average to a function value, rather than a derivative to a secant slope.
Cauchy's mean value theorem: A theorem relating the ratios of changes in two functions to the ratio of their derivatives at an interior point. It generalizes the theorem from one function to a pair of functions.
Lagrange's error bound: An estimate that bounds a Taylor polynomial’s error using a higher derivative on an interval. The bound follows by applying the mean value theorem to the Taylor remainder.
Differentiability: The property of having a derivative at a point or throughout a specified domain. Differentiability at every interior point is the theorem’s other essential hypothesis.
Darboux's theorem: A theorem stating that derivatives have the intermediate value property, even when they are not continuous. It constrains derivative values without asserting the secant-slope equality supplied here.
Intermediate value theorem: A theorem stating that a continuous function takes every value between its endpoint values. Continuity and connected intervals help explain why derivative values cannot skip the required average.
Monotonic function: A function that never decreases or never increases across its domain. Derivative sign tests for monotonicity follow from the theorem applied between any two points.

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