Mean value theorem

In mathematics, the mean value theorem states, roughly, that for a given a planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.

This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.

More precisely, if a function ${\displaystyle f}$ is continuous on the closed interval ${\displaystyle [a,b]}$ , and differentiable on the open interval ${\displaystyle (a,b)}$ , then there exists a point ${\displaystyle c}$ in ${\displaystyle (a,b)}$ such that:

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