 Mean value theorem

For the theorem in harmonic function theory, see Harmonic function#Mean value property.
In calculus, the mean value theorem states, roughly, that given an arc of a differentiable (continuous) curve, there is at least one point on that arc at which the derivative (slope) of the curve is equal to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the secant chord connecting the endpoints of the arc. The theorem is used to prove theorems that make global conclusions about a function on an interval starting from local hypotheses about derivatives at points of the interval.
More precisely, if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that
 ^{[1]}
This theorem can be understood intuitively by applying it to motion: If a car travels one hundred miles in one hour, then its average speed during that time was 100 miles per hour. To get at that average speed, the car either has to go at a constant 100 miles per hour during that whole time, or, if it goes slower at one moment, it has to go faster at another moment as well (and vice versa), in order to still end up with an average of 100 miles per hour. Therefore, the Mean Value Theorem tells us that at some point during the journey, the car must have been traveling at exactly 100 miles per hour; that is, it was traveling at its average speed.
A special case of this theorem was first described by Parameshvara (1370–1460) from the Kerala school of astronomy and mathematics in his commentaries on Govindasvāmi and Bhaskara II.^{[2]} The mean value theorem in its modern form was later stated by Augustin Louis Cauchy (1789–1857). 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 essential in proving the fundamental theorem of calculus. The mean value theorem follows from the more specific statement of Rolle's theorem, and can be used to prove the more general statement of Taylor's theorem (with Lagrange form of the remainder term).
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JacobianLet f : [a, b] → R be a continuous function on the closed interval [a, b], and differentiable on the open interval (a, b), where a < b. Then there exists some c in (a, b) such that
The mean value theorem is a generalization of Rolle's theorem, which assumes f(a) = f(b), so that the righthand side above is zero.
The mean value theorem is still valid in a slightly more general setting. One only needs to assume that f : [a, b] → R is continuous on [a, b], and that for every x in (a, b) the limit
exists as a finite number or equals +∞ or −∞. If finite, that limit equals f′(x). An example where this version of the theorem applies is given by the realvalued cube root function mapping x to x^{1/3}, whose derivative tends to infinity at the origin.
Note that the theorem is false if a differentiable function is complexvalued instead of realvalued. For example, define f(x) = e^{ix} for all real x. Then
 f(2π) − f(0) = 0 = 0(2π − 0)
while f′(x) = 1.
Proof
The expression (f(b) − f(a)) / (b − a) gives the slope of the line joining the points (a, f(a)) and (b, f(b)), which is a chord of the graph of f, while f′(x) gives the slope of the tangent to the curve at the point (x, f(x)). Thus the Mean value theorem says that given any chord of a smooth curve, we can find a point lying between the endpoints of the chord such that the tangent at that point is parallel to the chord. The following proof illustrates this idea.
Define g(x) = f(x) − rx, where r is a constant. Since f is continuous on [a, b] and differentiable on (a, b), the same is true for g. We now want to choose r so that g satisfies the conditions of Rolle's theorem. Namely
By Rolle's theorem, since g is continuous and g(a) = g(b), there is some c in (a, b) for which g'(c) = 0, and it follows from the equality g(x) = f(x) − rx that,
as required.
A simple application
Assume that f is a continuous, realvalued 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.
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:
 Only continuity of ƒ, 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.)
 The differentiability of ƒ can be relaxed to onesided differentiability, a proof given in the article on semidifferentiability.
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 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
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
The proof of Cauchy's mean value theorem is based on the same idea as the proof of the mean value theorem.
Define h(x) = f(x) − rg(x), where r is a constant. Since f and g are continuous on [a, b] and differentiable on (a, b), the same is true for h. We now want to choose r so that h satisfies the conditions of Rolle's theorem. Namely
By Rolle's theorem, since h is continuous and h(a) = h(b), there is some c in (a, b) for which h'(c) = 0, and it follows from the equality h(x) = f(x) − rg(x) that,
as required.
Mean value theorem in several variables
The mean value theorem in one variable generalizes to several variables by applying the theorem in one variable via parametrization. Let G be an open subset of R^{n}, and let f : G → R be a differentiable function. Fix points x, y ∈ G such that the interval x y lies in G, and define g(t) = f((1 − t)x + ty). Since g is a differentiable function in one variable, the mean value theorem gives:
for some c between 0 and 1. But since g(1) = f(y) and g(0) = f(x), computing g′(c) 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 n = 1 this is the theorem in one variable). By the Schwarz inequality, the equation gives the estimate:
In particular, when the partial derivatives of f are bounded, f is Lipschitz continuous (and therefore uniformly continuous). Note that f is not assumed to be continuously differentiable nor continuous on the closure of G. However, in the above, we used the chain rule so the existence of ∇f would not be sufficient.
As an application of the above, we prove that f is constant if G is connected and every partial derivative of f is 0. Pick some point x_{0} ∈ G, and let g(x) = f(x) − f(x_{0}). We want to show g(x) = 0 for every x ∈ G. For that, let E = {x ∈ G : g(x) = 0} . Then E is closed and nonempty. It is open too: for every x ∈ E,
for every y in some neighborhood of x. (Here, it is crucial that x and y are sufficiently close to each other.) Since G is connected, we conclude E = G.
Remark that all arguments in the above are made in a coordinatefree manner; hence, they actually generalize to the case when G is a subset of a Banach space.
Mean value theorem for vectorvalued functions
There is no exact analog of the mean value theorem for vectorvalued functions. The problem is roughly speaking the following: If is a differentiable function (where is open) and if is the line segment in question (lying inside ), then one can apply the above parametrization procedure to each of the component functions of (in the above notation set ). In doing so one finds points on the line segment satisfying
But generally there will not be a single point on the line segment satisfying
for all simultaneously. (As a counterexample one could take defined via the component functions . Then , but and are never simultaneously zero as ranges over .)
However a certain type of generalization of the mean value theorem to vectorvalued functions is obtained as follows: Let f be a continuously differentiable realvalued 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 between 0 and 1 such that
On the other hand we have
Thus, the value at the particular point has been replaced by the mean value . This last version can be generalized to vector valued functions:
Let be open, continuously differentiable, and vectors such that the whole line segment remains in . Then we have:
where the integral of a matrix is to be understood componentwise. (Dƒ denotes the Jacobian matrix of ƒ.)
From this one can further deduce that if Dƒ(x + th) is bounded for t between 0 and 1 by some constant M, then
Proof of (*). Write () for the real valued components of . Define the functions byThen we have
The claim follows since is the matrix consisting of the components , q.e.d.
Proof of (**). From (*) it follows that
Here we have used the following
Lemma. Let be a continuous function defined on the interval . Then we have
Proof of (***). Let denote the value of the integral Now
thus as desired. (Note the use of the Cauchy–Schwarz inequality.) This shows (***) and thereby finishes the proof of (**).
Mean value theorems for integration
First mean value theorem for integration
The first mean value theorem for integration states
 If is a continuous function and φ is an integrable function that does not change sign on the interval (a,b), then there exists a number such that
In particular, if φ(t) = 1 for all , then there exists such that
The point G(x) is called the mean value of G(t) on [a,b].
Proof of the first mean value theorem for integration
It follows from the extreme value theorem that the continuous function G has a finite infimum m and a finite supremum M on the interval [a, b]. From the monotonicity of the integral and the fact that m ≤ G(t) ≤ M, it follows that
where
denotes the integral of φ. Hence, if I = 0, then the claimed equality holds for every x in [a, b]. Therefore, we may assume I > 0 in the following. Dividing through by I we have that
The extreme value theorem tells us more than just that the infimum and supremum of G on [a, b] are finite; it tells us that both are actually attained. Thus we can apply the intermediate value theorem, and conclude that the continuous function G attains every value of the interval [m, M], in particular there exists x in [a, b] such that
This completes the proof.
Second mean value theorem for integration
There are various slightly different theorems called the second mean value theorem for integration. 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 G(a + 0) 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:
 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
This variant was proved by Hiroshi Okamura in 1947.^{[citation needed]}
A probabilistic analogue of the mean value theorem
Let X and Y be nonnegative random variables such that and (i.e. X is smaller than Y in the usual stochastic order). Then there exists an absolutely continuous nonnegative random variable Z having probability density function
Let g be a measurable and differentiable function such that E[g(X)] and E[g(Y)] are finite, and let its derivative g' be measurable and Riemannintegrable on the interval [x,y] for all . Then, E[g'(Z)] is finite and^{[3]}
See also
Notes
 ^ Weisstein, Eric. "MeanValue Theorem". MathWorld. Wolfram Research. http://mathworld.wolfram.com/MeanValueTheorem.html. Retrieved 24 March 2011.
 ^ J. J. O'Connor and E. F. Robertson (2000). Paramesvara, MacTutor History of Mathematics archive.
 ^ A. Di Crescenzo (1999). A probabilistic analogue of the mean value theorem and its applications to reliability theory. J. Appl. Prob. 36, 706719.
External links
Categories: Calculus
 Theorems in calculus
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