In mathematics, the mean value theorem (or Lagrange's mean value 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.

A special case of this theorem for inverse interpolation of the sine was first described by Parameshvara (1380–1460), from the Kerala School of Astronomy and Mathematics in India, in his commentaries on Govindasvāmi and Bhāskara II. A restricted form of the theorem was proved by Michel Rolle in 1691; the result was what is now known as Rolle's theorem, and was proved only for polynomials, without the techniques of calculus. The mean value theorem in its modern form was stated and proved by Augustin Louis Cauchy in 1823. Many variations of this theorem have been proved since then.

Let ${\displaystyle f:[a,b]\to \mathbb {R} }$ be a continuous function on the closed interval ${\displaystyle [a,b]}$, and differentiable on the open interval ${\displaystyle (a,b)}$, where ${\displaystyle a<b}$. Then there exists some ${\displaystyle c}$ in ${\displaystyle (a,b)}$ such that:

The mean value theorem is a generalization of Rolle's theorem, which assumes ${\displaystyle f(a)=f(b)}$, so that the right-hand side above is zero.

The mean value theorem is still valid in a slightly more general setting. One only needs to assume that ${\displaystyle f:[a,b]\to \mathbb {R} }$ is continuous on ${\displaystyle [a,b]}$, and that for every ${\displaystyle x}$ in ${\displaystyle (a,b)}$ the limit

exists as a finite number or equals ${\displaystyle \infty }$ or ${\displaystyle -\infty }$. If finite, that limit equals ${\displaystyle f'(x)}$. An example where this version of the theorem applies is given by the real-valued cube root function mapping ${\displaystyle x\mapsto x^{1/3}}$, whose derivative tends to infinity at the origin.

The expression ${\textstyle {\frac {f(b)-f(a)}{b-a}}}$ gives the slope of the line joining the points ${\displaystyle (a,f(a))}$ and ${\displaystyle (b,f(b))}$, which is a chord of the graph of ${\displaystyle f}$, while ${\displaystyle f'(x)}$ gives the slope of the tangent to the curve at the point ${\displaystyle (x,f(x))}$. Thus the mean value theorem says that given any chord of a smooth curve, we can find a point on the curve lying between the end-points of the chord such that the tangent of the curve at that point is parallel to the chord. The following proof illustrates this idea.

Define ${\displaystyle g(x)=f(x)-rx}$, where ${\displaystyle r}$ is a constant. Since ${\displaystyle f}$ is continuous on ${\displaystyle [a,b]}$ and differentiable on ${\displaystyle (a,b)}$, the same is true for ${\displaystyle g}$. We now want to choose ${\displaystyle r}$ so that ${\displaystyle g}$ satisfies the conditions of Rolle's theorem. Namely