Mean value theorem

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.

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.

The mean value theorem is still valid in a slightly more general setting.

the limit exists as a finite number or equals

An example where this version of the theorem applies is given by the real-valued cube root function mapping

gives the slope of the line joining the points

gives the slope of the tangent to the curve at the point

is a continuous, real-valued function, defined on an arbitrary interval

, this is equivalent to: Geometrically, this means that there is some tangent to the graph of the curve[8] which is parallel to the line defined by the points

However, Cauchy's theorem does not claim the existence of such a tangent in all cases where

, yet never has a horizontal tangent; however it has a stationary point (in fact a cusp) at

Cauchy's mean value theorem can be used to prove L'Hôpital's rule.

The mean value theorem generalizes to real functions of multiple variables.

The trick is to use parametrization to create a real function of one variable, and then apply the one-variable theorem.

is a differentiable function in one variable, the mean value theorem gives: for some

This is an exact analog of the theorem in one variable (in the case

The above arguments are made in a coordinate-free manner; hence, they generalize to the case when

There is no exact analog of the mean value theorem for vector-valued functions (see below).

Serge Lang in Analysis I uses the mean value theorem, in integral form, as an instant reflex but this use requires the continuity of the derivative.

The reason why there is no analog of mean value equality is the following: If f : U → Rm is a differentiable function (where U ⊂ Rn is open) and if x + th, x, h ∈ Rn, t ∈ [0, 1] is the line segment in question (lying inside U), then one can apply the above parametrization procedure to each of the component functions fi (i = 1, …, m) of f (in the above notation set y = x + h).

In doing so one finds points x + tih on the line segment satisfying But generally there will not be a single point x + t*h on the line segment satisfying for all i simultaneously.

The above theorem implies the following: Mean value inequality[10] — For a continuous function

, then In fact, the above statement suffices for many applications and can be proved directly as follows.

The theorem is false if a differentiable function is complex-valued instead of real-valued.

[12] In general, if f : [a, b] → R is continuous and g is an integrable function that does not change sign on [a, b], then there exists c in (a, b) such that There are various slightly different theorems called the second mean value theorem for definite integrals.

returns a multi-dimensional vector, then the MVT for integration is not true, even if the domain of

Then there exists an absolutely continuous non-negative random variable Z having probability density function Let g be a measurable and differentiable function such that E[g(X)], E[g(Y)] < ∞, and let its derivative g′ be measurable and Riemann-integrable on the interval [x, y] for all y ≥ x ≥ 0.

Then, E[g′(Z)] is finite and[14] As noted above, the theorem does not hold for differentiable complex-valued functions.