Mean value theorem

In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given 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. It is one of the most important results in real analysis. 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, the theorem states that if ${\displaystyle f}$ is a continuous function 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 the tangent at ${\displaystyle c}$ is parallel to the secant line through the endpoints ${\displaystyle {\big (}a,f(a){\big )}}$ and ${\displaystyle {\big (}b,f(b){\big )}}$, that is,

${\displaystyle f'(c)={\frac {f(b)-f(a)}{b-a}}.}$