Mean value theorem

The mean value theorem is one of the most important theorems of calculus and analysis. It states, roughly, that a continuous function differentiable over some interval always contains at least one point where the tangent line is parallel to the secant line spanning the interval.[cite:@openstax2020calculus]

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The theorem can be intuited as a "skewing" of Rolle's theorem.

Statement

Let $f$ be a function that is continuous over the closed interval $\text{intCC}ab$ and differentiable over the open interval $\text{intOO}ab$. Then there exists a value $c\in \text{intOO}ab$ such that the line tangent to $f$ at $c$ is parallel to the secant line connecting the points $(a,fa)$ and $(b,fb)$; or, symbolically,[cite:@openstax2020calculus] $$f^{\prime }c=\frac{fb-fa}{b-a}.$$