For any function that is continuous on and differentiable on there exists some in the interval such that the secant joining the endpoints of the interval is parallel to the tangent at .
In mathematics, the 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.
The mean value theorem is a generalization of Rolle's theorem, which assumes , 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 is continuous on , and that for every in the limit
exists as a finite number or equals or . If finite, that limit equals . An example where this version of the theorem applies is given by the real-valued cube root function mapping , whose derivative tends to infinity at the origin.
Note that the theorem, as stated, is false if a differentiable function is complex-valued instead of real-valued. For example, define for all real . Then
while for any real .
These formal statements are also known as Lagrange's Mean Value Theorem.
The expression gives the slope of the line joining the points and , which is a chord of the graph of , while gives the slope of the tangent to the curve at the point . 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 , where is a constant. Since is continuous on and differentiable on , the same is true for . We now want to choose so that satisfies the conditions of Rolle's theorem. Namely
By Rolle's theorem, since is differentiable and , there is some in for which , and it follows from the equality that,
Theorem 1: Assume that f is a continuous, real-valued function, defined on an arbitrary interval I of the real line. If the derivative of f at every interior point of the interval I exists and is zero, then f is constant in the interior.
Proof: Assume the derivative of f at every interior point of the interval I exists and is zero. Let (a, b) be an arbitrary open interval in I. By the mean value theorem, there exists a point c in (a,b) such that
This implies that f(a) = f(b). Thus, f is constant on the interior of I and thus is constant on I by continuity. (See below for a multivariable version of this result.)
Only continuity of f, not differentiability, is needed at the endpoints of the interval I. No hypothesis of continuity needs to be stated if I is an open interval, since the existence of a derivative at a point implies the continuity at this point. (See the section continuity and differentiability of the article derivative.)
Theorem 2: If f(x) = g(x) for all x in an interval (a, b) of the domain of these functions, then f - g is constant, i.e. f = g + c where c is a constant on (a, b).
Proof: Let F = f − g, then F' = f' − g' = 0 on the interval (a, b), so the above theorem 1 tells that F = f − g is a constant c or f = g + c.
Theorem 3: If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F(x) + c where c is an constant.
Proof: It directly follows from the theorem 2 above.
Cauchy's mean value theorem
Cauchy's mean value theorem, also known as the extended mean value theorem, is a generalization of the mean value theorem. It states: if the functions and are both continuous on the closed interval and differentiable on the open interval , then there exists some , such that
Geometrical meaning of Cauchy's theorem
Of course, if and , this is equivalent to:
Geometrically, this means that there is some tangent to the graph of the curve
which is parallel to the line defined by the points and . However, Cauchy's theorem does not claim the existence of such a tangent in all cases where and are distinct points, since it might be satisfied only for some value with , in other words a value for which the mentioned curve is stationary; in such points no tangent to the curve is likely to be defined at all. An example of this situation is the curve given by
which on the interval goes from the point to , 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 is the special case of Cauchy's mean value theorem when .
Proof of Cauchy's mean value theorem
The proof of Cauchy's mean value theorem is based on the same idea as the proof of the mean value theorem.
Suppose . Define , where is fixed in such a way that , namely
Since and are continuous on and differentiable on , the same is true for . All in all, satisfies the conditions of Rolle's theorem: consequently, there is some in for which . Now using the definition of we have:
If , then, applying Rolle's theorem to , it follows that there exists in for which . Using this choice of , Cauchy's mean value theorem (trivially) holds.
Generalization for determinants
Assume that and are differentiable functions on that are continuous on . Define
There exists such that .
and if we place , we get Cauchy's mean value theorem. If we place and we get Lagrange's mean value theorem.
The proof of the generalization is quite simple: each of and are determinants with two identical rows, hence . The Rolle's theorem implies that there exists such that .
Mean value theorem in several variables
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.
Let be an open convex subset of , and let be a differentiable function. Fix points , and define . Since is a differentiable function in one variable, the mean value theorem gives:
for some between 0 and 1. But since and , computing explicitly we have:
where denotes a gradient and a dot product. Note that this is an exact analog of the theorem in one variable (in the case this is the theorem in one variable). By the Cauchy–Schwarz inequality, the equation gives the estimate:
As an application of the above, we prove that is constant if is open and connected and every partial derivative of is 0. Pick some point , and let . We want to show for every . For that, let . Then E is closed and nonempty. It is open too: for every ,
for every in some neighborhood of . (Here, it is crucial that and are sufficiently close to each other.) Since is connected, we conclude .
The above arguments are made in a coordinate-free manner; hence, they generalize to the case when is a subset of a Banach space.
Mean value theorem for vector-valued functions
There is no exact analog of the mean value theorem for vector-valued functions.
In Principles of Mathematical Analysis,Rudin gives an inequality which can be applied to many of the same situations to which the mean value theorem is applicable in the one dimensional case:
Theorem — For a continuous vector-valued function differentiable on , there exists such that .
Jean Dieudonné in his classic treatise Foundations of Modern Analysis discards the mean value theorem and replaces it by mean inequality as the proof is not constructive and one cannot find the mean value and in applications one only needs mean inequality. 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. If one uses the Henstock–Kurzweil integral one can have the mean value theorem in integral form without the additional assumption that derivative should be continuous as every derivative is Henstock–Kurzweil integrable. The problem is roughly speaking 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 isimultaneously. For example, define:
Then , but and are never simultaneously zero as ranges over .
However a certain type of generalization of the mean value theorem to vector-valued functions is obtained as follows: Let f be a continuously differentiable real-valued function defined on an open interval I, and let x as well as x + h be points of I. The mean value theorem in one variable tells us that there exists some t* between 0 and 1 such that
Now cancelling the norm of u from both ends gives us the desired inequality.
Mean Value Inequality — If the norm of Df(x + th) is bounded by some constant M for t in [0, 1], then
Proof. From Lemma 1 and 2 it follows that
Mean value theorems for definite integrals
First mean value theorem for definite integrals
Geometrically: interpreting f(c) as the height of a rectangle and b–a as the width, this rectangle has the same area as the region below the curve from a to b
Let f : [a, b] → R be a continuous function. Then there exists c in [a, b] such that
Since the mean value of f on [a, b] is defined as
we can interpret the conclusion as f achieves its mean value at some c in (a, b).
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
Proof of the first mean value theorem for definite integrals
Suppose f : [a, b] → R is continuous and g is a nonnegative integrable function on [a, b]. By the extreme value theorem, there exists m and M such that for each x in [a, b], and . Since g is nonnegative,
^Ádám Besenyei. "Historical development of the mean value theorem" (PDF).
^Lozada-Cruz, German (2020-10-02). "Some variants of Cauchy's mean value theorem". International Journal of Mathematical Education in Science and Technology. 51 (7): 1155–1163. doi:10.1080/0020739X.2019.1703150. ISSN 0020-739X.
^Sahoo, Prasanna. (1998). Mean value theorems and functional equations. Riedel, T. (Thomas), 1962-. Singapore: World Scientific. ISBN 981-02-3544-5. OCLC 40951137.
^ abcKirshna's Real Analysis: (General). Krishna Prakashan Media.