Mean value theorem

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For the theorem in harmonic function theory, see Harmonic function § The mean value property.
Not to be confused with the Intermediate value theorem.
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 given a 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.

The theorem is used to prove global statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.

More precisely, if a function is continuous on the closed interval , where , and differentiable on the open interval , then there exists a point in such that:[1]

It is one of the most important results in differential calculus, as well as one of the most important theorems in mathematical analysis, and is useful in proving the fundamental theorem of calculus. The mean value theorem follows from a more specific statement of Rolle's theorem, and can be used to prove the most general statement of Taylor's theorem (with Lagrange form of the remainder term).

History[edit]

A special case of this theorem was first described by Parameshvara (1370–1460), from the Kerala school of astronomy and mathematics in India, in his commentaries on Govindasvāmi and Bhaskara II.[2] A restricted form of the theorem was proved by 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 Cauchy in 1823.[3]

Formal statement[edit]

Let be a continuous function on the closed interval , and differentiable on the open interval , where . Then there exists some in such that

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 .

Proof[edit]

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 lying between the end-points of the chord such that the tangent 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,

as required.

A simple application[edit]

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.)

Remarks:

Cauchy's mean value theorem[edit]

Cauchy's mean value theorem, also known as the extended mean value theorem, is a generalization of the mean value theorem. It states: If functions f and g are both continuous on the closed interval [a, b], and differentiable on the open interval (a, b), then there exists some c ∈ (a, b), such that

Geometrical meaning of Cauchy's theorem

Of course, if g(a) ≠ g(b) and if g′(c) ≠ 0, 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 (f(a), g(a)) and (f(b), g(b)). However Cauchy's theorem does not claim the existence of such a tangent in all cases where (f(a), g(a)) and (f(b), g(b)) are distinct points, since it might be satisfied only for some value c with f′(c) = g′(c) = 0, 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 [−1, 1] goes from the point (−1, 0) to (1, 0), yet never has a horizontal tangent; however it has a stationary point (in fact a cusp) at t = 0.

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 g(t) = t.

Proof of Cauchy's mean value theorem[edit]

The proof of Cauchy's mean value theorem is based on the same idea as the proof of the mean value theorem.

  • Suppose g(a) ≠ g(b). Define h(x) = f(x) − rg(x), where r is fixed in such a way that h(a) = h(b), namely
Since f and g are continuous on [a, b] and differentiable on (a, b), the same is true for h. All in all, h satisfies the conditions of Rolle's theorem: consequently, there is some c in (a, b) for which h′(c) = 0. Now using the definition of h we have:
Therefore:
which implies the result.
  • If g(a) = g(b), then, applying Rolle's theorem to g, it follows that there exists c in (a, b) for which g′(c) = 0. Using this choice of c, Cauchy's mean value theorem (trivially) holds.

Generalization for determinants[edit]

Assume that and are differentiable functions on that are continuous on . Define

There exists such that .

Notice 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[edit]

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 connected subset of , and let be a differentiable function. Fix points such that the interval lies in , 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:

In particular, when the partial derivatives of are bounded, is Lipschitz continuous (and therefore uniformly continuous). Note that is not assumed to be continuously differentiable nor continuous on the closure of . However, in the above, we used the chain rule so the existence of would not be sufficient.

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 .

Remark that all arguments in the above are made in a coordinate-free manner; hence, they actually generalize to the case when is a subset of a Banach space.

Mean value theorem for vector-valued functions[edit]

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

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 : URm is a differentiable function (where URn is open) and if x + th, x, hRn, 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. 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

On the other hand, we have, by the fundamental theorem of calculus followed by a change of variables,

Thus, the value f′(x + t*h) at the particular point t* has been replaced by the mean value

This last version can be generalized to vector valued functions:

Lemma 1. Let URn be open, f : URm continuously differentiable, and xU, hRn vectors such that the line segment x + th, 0 ≤ t ≤ 1 remains in U. Then we have:
where Df denotes the Jacobian matrix of f and the integral of a matrix is to be understood componentwise.

Proof. Let f1, ..., fm denote the components of f and define:

Then we have

The claim follows since Df is the matrix consisting of the components

Lemma 2. Let v : [a, b] → Rm be a continuous function defined on the interval [a, b] ⊂ R. Then we have

Proof. Let u in Rm denote the value of the integral

Now we have (using the Cauchy–Schwarz inequality):

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[edit]

First mean value theorem for definite integrals[edit]

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).[4]

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[edit]

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,

Now let

If , we're done since

means

so for any c in (a, b),

If I ≠ 0, then

By the intermediate value theorem, f attains every value of the interval [m, M], so for some c in [a, b]

that is,

Finally, if g is negative on [a, b], then

and we still get the same result as above.

QED

Second mean value theorem for definite integrals[edit]

There are various slightly different theorems called the second mean value theorem for definite integrals. A commonly found version is as follows:

If G : [a, b] → R is a positive monotonically decreasing function and φ : [a, b] → R is an integrable function, then there exists a number x in (a, b] such that

Here stands for , the existence of which follows from the conditions. Note that it is essential that the interval (a, b] contains b. A variant not having this requirement is:[5]

If G : [a, b] → R is a monotonic (not necessarily decreasing and positive) function and φ : [a, b] → R is an integrable function, then there exists a number x in (a, b) such that

Mean value theorem for integration fails for vector-valued functions[edit]

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

For example, consider the following 2-dimensional function defined on an -dimensional cube:

Then, by symmetry it is easy to see that the mean value of over its domain is (0,0):

However, there is no point in which , because everywhere.

A probabilistic analogue of the mean value theorem[edit]

Let X and Y be non-negative random variables such that E[X] < E[Y] < ∞ and (i.e. X is smaller than Y in the usual stochastic order). 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 yx ≥ 0. Then, E[g′(Z)] is finite and[6]

Generalization in complex analysis[edit]

As noted above, the theorem does not hold for differentiable complex-valued functions. Instead, a generalization of the theorem is stated such:[7]

Let f : Ω → C be a holomorphic function on the open convex set Ω, and let a and b be distinct points in Ω. Then there exist points u, v on Lab (the line segment from a to b) such that

Where Re() is the Real part and Im() is the Imaginary part of a complex-valued function.

See also[edit]

Notes[edit]

  1. ^ Weisstein, Eric. "Mean-Value Theorem". MathWorld. Wolfram Research. Retrieved 24 March 2011. 
  2. ^ J. J. O'Connor and E. F. Robertson (2000). Paramesvara, MacTutor History of Mathematics archive.
  3. ^ A. Besenyei, Historical development of the mean value theorem, http://abesenyei.web.elte.hu/publications/meanvalue.pdf
  4. ^ Michael Comenetz (2002). Calculus: The Elements. World Scientific. p. 159. ISBN 978-981-02-4904-5. 
  5. ^ Hobson, E. W. (1909). "On the Second Mean-Value Theorem of the Integral Calculus". Proc. London Math Soc. S2–7 (1): 14–23. doi:10.1112/plms/s2-7.1.14. MR 1575669. 
  6. ^ Di Crescenzo, A. (1999). "A Probabilistic Analogue of the Mean Value Theorem and Its Applications to Reliability Theory". J. Appl. Prob. 36 (3): 706–719. JSTOR 3215435. 
  7. ^ "Complex Mean-Value Theorem". PlanetMath. PlanetMath. 

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