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 ${\displaystyle }$ 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,

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

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.^[1] 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.^[2] Many variations of this theorem have been proved since then.^[3][4]

Let ${\displaystyle f:\to \mathbb {R} }$ be a continuous function on the closed interval ${\displaystyle }$, 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

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

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

${\displaystyle \lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}}$

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 theorem, as stated, is false if a differentiable function is complex-valued instead of real-valued. For example, define $$Zdroj:https://en.wikipedia.org?pojem=Mean_value_theorem
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