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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

to the arc is parallel to the secant through its endpoints. It is one of the most important results in

real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include con ...

. 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

f

is a continuous function on the

closed interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...

, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline of ...

/math> and

differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point i ...

on the open interval

(a,b)

, then there exists a point

c

(a,b)

such that the tangent at

c

is parallel to the secant line through the endpoints

\big(a, f(a)\big)

and

\big(b, f(b)\big)

, that is, :

f'(c)=\frac.

History

A special case of this

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...

for inverse interpolation of the sine was first described by

Parameshvara Vatasseri Parameshvara Nambudiri ( 1380–1460) was a major Indian mathematician and astronomer of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama. He was also an astrologer. Parameshvara was a proponent of ...

(1380–1460), from the

Kerala School of Astronomy and Mathematics The Kerala school of astronomy and mathematics or the Kerala school was a school of Indian mathematics, mathematics and Indian astronomy, astronomy founded by Madhava of Sangamagrama in Kingdom of Tanur, Tirur, Malappuram district, Malappuram, K ...

India India, officially the Republic of India ( Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the ...

, in his commentaries on

Govindasvāmi Govindasvāmi (or Govindasvāmin, Govindaswami) (c. 800 – c. 860) was an Indian mathematical astronomer most famous for his ''Bhashya'', a commentary on the ''Mahābhāskarīya'' of Bhāskara I, written around 830. The commentary contains many ...

and

Bhāskara II Bhāskara II (c. 1114–1185), also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian mathematician and astronomer. From verses, in his main work, Siddhānta Shirom ...

. A restricted form of the theorem was proved by

Michel Rolle Michel Rolle (21 April 1652 – 8 November 1719) was a French mathematician. He is best known for Rolle's theorem (1691). He is also the co-inventor in Europe of Gaussian elimination (1690). Life Rolle was born in Ambert, Basse-Auvergne. Rol ...

in 1691; the result was what is now known as

Rolle's theorem In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point wher ...

, 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 Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...

in 1823. Many variations of this theorem have been proved since then.

Formal statement

Let

to\R

be a continuous function on the closed interval and

on the open interval where Then there exists some

c

(a,b)

such that :

f'(c)=\frac.

The mean value theorem is a generalization of

, which assumes

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

to\R

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

,b /math>, and that for every x in (a,b) the

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

\lim_\frac

exists as a finite number or equals

\infty

-\infty

. If finite, that limit equals

f'(x)

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

x \mapsto x^

, whose

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

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

f(x) = e^

for all real Then :

f(2\pi)-f(0)=0=0(2\pi-0)

while

f'(x)\ne 0

for any real These formal statements are also known as

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia

Proof

The expression

\frac

gives the

slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used ...

of the line joining the points

(a,f(a))

and

(b,f(b))

, which is a

chord Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * 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 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

g(x)=f(x)-rx

, where

r

is a constant. Since

f

is continuous on

,b /math> and differentiable on (a,b), the same is true for g . We now want to choose r so that g satisfies the conditions of

. Namely :

\begin
g(a)=g(b)&\iff f(a)-ra=f(b)-rb\\
&\iff r(b-a)=f(b)-f(a) \\
&\iff r=\frac .
\end

, since

g

is differentiable and

g(a)=g(b)

, there is some

c

(a,b)

for which

g'(c)=0

, and it follows from the equality

g(x)=f(x)-rx

that, :

\begin
&g'(x) = f'(x) -r \\
& g'(c) = 0\\
&g'(c) = f'(c) - r = 0 \\
&\Rightarrow f'(c) = r = \frac
\end

Implications

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 In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of t ...

of the interval ''I'' exists and is zero, then ''f'' is

constant Constant or The Constant may refer to: Mathematics * Constant (mathematics), a non-varying value * Mathematical constant, a special number that arises naturally in mathematics, such as or Other concepts * Control variable or scientific const ...

in the interior. Proof: Assume the derivative of ''f'' at every

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 :

0=f'(c)=\frac.

This implies that . 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 ''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

.) * The differentiability of ''f'' can be relaxed to one-sided differentiability, a proof given in the article on

semi-differentiability In calculus, a branch of mathematics, the notions of one-sided differentiability and semi-differentiability of a real-valued function ''f'' of a real variable are weaker than differentiability. Specifically, the function ''f'' is said to be righ ...

. 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 a 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

f

and

g

are both continuous on the closed interval

,b /math> and differentiable on the open interval (a,b), then there exists some c \in (a,b), such that : (f(b)-f(a))g'(c)=(g(b)-g(a))f'(c). Of course, if g(a) \neq g(b) and g'(c) \neq 0, this is equivalent to:

: \frac=\frac. Geometrically, this means that there is some

to the graph of the

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

\to \R^2\\t\mapsto (f(t),g(t))\end

which is

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of I ...

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 :

t \mapsto \left(t^3,1-t^2\right),

which on the interval

1,1 /math> 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 A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifurca ...

) at

t = 0

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

L'Hôpital's rule In calculus, l'Hôpital's rule or l'Hospital's rule (, , ), also known as Bernoulli's rule, is a theorem which provides a technique to evaluate limits of indeterminate forms. Application (or repeated application) of the rule often converts an ...

. 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.

Suppose $g(a) \neq g(b)$ . Define $h(x) = f(x) - rg(x)$ , where $r$ is fixed in such a way that $h(a) = h(b)$ , namely : $\beginh(a)=h(b)&\iff f(a)-rg(a)=f(b)-rg(b)\\ &\iff r (g(b)-g(a))=f(b)-f(a)\\ &\iff r=\frac.\end$ Since $f$ and $g$ are continuous on $,b /math> and differentiable on (a,b), the same is true for h . All in all, h satisfies the conditions of$
Rolle's theorem In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point wher ...
: consequently, there is some $c$ in $(a,b)$ for which $h'(c) = 0$ . Now using the definition of $h$ we have: : $0=h'(c)=f'(c)-rg'(c) = f'(c)- \left (\frac \right ) g'(c).$ Therefore: : $f'(c)= \frac g'(c),$ which implies the result.
If $g(a) = g(b)$ , then, applying
Rolle's theorem In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point wher ...
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

Assume that

f, g,

and

h

are differentiable functions on

(a,b)

that are continuous on

,b /math>. Define

: D(x) = \begin
f(x) & g(x) & h(x)\\
f(a) & g(a) & h(a)\\
f(b) & g(b) & h(b)
\end There exists c\in(a,b) such that D'(c)=0 .

Notice that
: D'(x) = \begin
f'(x) & g'(x)& h'(x)\\
f(a) & g(a) & h(a)\\
f(b) & g(b)& h(b)
\end and if we place h(x)=1, we get Cauchy's mean value theorem. If we place h(x)=1 and g(x)=x we get Lagrange's mean value theorem.

The proof of the generalization is quite simple: each of D(a) and D(b) are determinants with two identical rows, hence D(a)=D(b)=0 . The Rolle's theorem implies that there exists c\in (a,b) such that D'(c)=0 .

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

G

be an open subset of

\R^n

, and let

f:G\to\R

be a differentiable function. Fix points

x,y\in G

such that the line segment between

x, y

lies in

G

, and define

g(t)=f\big((1-t)x+ty\big)

. Since

g

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

g(1)-g(0)=g'(c)

for some

c

between 0 and 1. But since

g(1)=f(y)

and

g(0)=f(x)

, computing

g'(c)

explicitly we have: :

f(y)-f(x)=\nabla f\big((1-c)x+cy\big)\cdot (y-x)

where

\nabla

denotes a

gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...

and

\cdot

dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...

. 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

Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality f ...

, the equation gives the estimate: :

\Bigl, f(y)-f(x)\Bigr,  \le \Bigl, \nabla f\big((1-c)x+cy\big)\Bigr, \ \Bigl, y - x\Bigr, .

In particular, when the partial derivatives of

f

are bounded,

f

Lipschitz continuous In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there ...

(and therefore

uniformly continuous In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. I ...

). As an application of the above, we prove that

f

is constant if the open subset

G

is connected and every partial derivative of

f

is 0. Pick some point

x_0\in G

, and let

g(x)=f(x)-f(x_0)

. We want to show

g(x)=0

for every

x\in G

. For that, let

E=\

. Then ''E'' is closed and nonempty. It is open too: for every

x\in E

, :

\Big, g(y)\Big, =\Big, g(y)-g(x)\Big, \le (0)\Big, y-x\Big, =0

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

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

G

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 (see below). However, there is 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: The theorem follows from the mean value theorem. Indeed, take

\varphi(t) = (\textbf(b) - \textbf(a)) \cdot \textbf(t)

. Then

\varphi

is real-valued and thus, by the mean value theorem, :

\varphi(b) - \varphi(a) = \varphi'(c)(b-a)

for some

c \in (a, b)

. Now,

\varphi(b) - \varphi(a) = , \textbf(b) - \textbf(a), ^2

and

\varphi'(c) = (\textbf(b) - \textbf(a)) \cdot \textbf'(c).

Hence, using the

, from the above equation, we get: :

, \textbf(b) - \textbf(a), ^2 \le , \textbf(b) - \textbf(a),  , \textbf'(c) , (b-a).

\textbf(b) = \textbf(a)

, the theorem is trivial (any ''c'' works). Otherwise, dividing both sides by

, \textbf(b) - \textbf(a),

yields the theorem.

\square

Jean Dieudonné in his classic treatise ''Foundations of Modern Analysis'' discards the mean value theorem and replaces it by mean inequality (which is given below) as the proof is not constructive and one cannot find the mean value and in applications one only needs mean inequality.

Serge Lang Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the i ...

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 In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced ), Luzin integral or Perron integral, but not to be confused with the more general wide ...

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 reason why there is no analog of mean value equality is 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 ''f'' (in the above notation set ). In doing so one finds points on the line segment satisfying :

f_i(x+h) - f_i(x) = \nabla f_i (x + t_ih) \cdot h.

But generally there will not be a ''single'' point on the line segment satisfying :

f_i(x+h) - f_i(x) = \nabla f_i (x + t^* h) \cdot h.

for all ''simultaneously''. For example, define: :

\to \R^2 \\ f(x) = (\cos(x), \sin(x)) \end

Then

f(2\pi) - f(0) = \mathbf \in \R^2

, but

f_1'(x) = -\sin (x)

and

f_2'(x) = \cos (x)

are never simultaneously zero as

x

ranges over

\left, 2 \pi\right /math>.

The above theorem implies the following:


In fact, the above statement suffices for many applications and can be proved directly as follows. (We shall write f for \textbf for readability.) First assume f is differentiable at a too. If f' is unbounded on (a, b), there is nothing to prove. Thus, assume \sup_ , f',  < \infty . Let M > \sup_ , f', be some real number. Let
: E = \. We want to show 1 \in E . By continuity of f, the set E is closed. It is also nonempty as 0 is in it. Hence, the set E has the largest element s . If s = 1, then 1 \in E and we are done. Thus suppose otherwise. For 1 > t > s,
: \begin
&, f(a + t(b-a)) - f(a),  \\
&\le , f(a + t(b-a)) - f(a+s(b - a)) - f'(a + s(b-a))(t-s)(b-a),  + , f'(a+s(b-a)), (t-s)(b-a) \\
&+, f(a + s(b-a)) - f(a), .
\end Let \epsilon > 0 be such that M - \epsilon > \sup_ , f', . By the differentiability of f at a + s(b-a) (note s may be 0), if t is sufficiently close to s, the first term is \le \epsilon (t-s)(b-a) . The second term is \le (M - \epsilon) (t-s)(b-a) . The third term is \le Ms(b-a) . Hence, summing the estimates up, we get:, f(a + t(b-a)) - f(a),  \le tM, b-a,, a contradiction to the maximality of s . Hence, 1 = s \in M and that means:
:, f(b) - f(a),  \le M(b-a). Since M is arbitrary, this then implies the assertion. Finally, if f is not differentiable at a, let a' \in (a, b) and apply the first case to f restricted on', b /math>, giving us:
:, f(b) - f(a'),  \le (b-a')\sup_ , f', since (a', b) \subset (a, b) . Letting a' \to a finishes the proof. \square For some applications of mean value inequality to establish basic results in calculus, see also Calculus on Euclidean space#Basic notions .

A certain type of generalization of the mean value theorem to vector-valued functions is obtained as follows: Let  be a continuously differentiable real-valued function defined on an open interval , and let  as well as  be points of . The mean value theorem in one variable tells us that there exists some  between 0 and 1 such that

: f(x+h)-f(x) = f'(x + t^* h) \cdot h. On the other hand, we have, by the

fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...

followed by a change of variables, :

f(x+h)-f(x) = \int_x^ f'(u) \, du = \left (\int_0^1 f'(x+th)\,dt \right) \cdot h.

Thus, the value at the particular point has been replaced by the mean value :

\int_0^1 f'(x + t h) \, dt.

This last version can be generalized to vector valued functions: Proof. Let ''f''₁, …, ''f_m'' denote the components of and define: :

\to \R \\ g_i(t) = f_i (x +th) \end

Then we have :

\begin
f_i(x+h)-f_i(x)
&= g_i(1)-g_i(0)
=\int_0^1 g_i'(t)\,dt \\
&= \int_0^1 \left (\sum_^n \frac (x + t h) h_j\right) dt
= \sum_^n \left (\int_0^1 \frac(x + t h)\,dt\right) h_j.
\end

The claim follows since is the matrix consisting of the components

\tfrac

\square

The mean value inequality can then be obtained as a corollary of the above proposition (though under the assumption the derivatives are continuous).

Cases where theorem cannot be applied (Necessity of conditions)

Both conditions for Mean Value Theorem are necessary: # f(x) is differentiable on (a,b) # f(x) is continuous on ,b'' Where one of the above conditions is not satisfied, Mean Value Theorem is not valid in general, and so it cannot be applied. Function is differentiable on open interval a,b The necessity of the first condition can be seen by the counterexample where the function

f(x)=, x,

on 1,1is not differentiable. Function is continuous on closed interval a,b The necessity of the second condition can be seen by the counterexample where the function

f(x) = \begin 1, & \textx=0 \\ 0, & \textx\in( 0,1] \end

f(x)

satisfies criteria 1 since

f'(x)=0

(0,1)

But not criteria 2 since

\frac=-1

and

-1\neq 0 =f'(x)

for all

x\in (0,1)

so no such

c

exists

Mean value theorems for definite integrals

First mean value theorem for definite integrals

Let ''f'' : 'a'', ''b''→ R be a continuous function. Then there exists ''c'' in (''a'', ''b'') such that :

\int_a^b f(x) \, dx = f(c)(b - a).

Since the mean value of ''f'' on 'a'', ''b''is defined as :

\frac \int_a^b f(x) \, dx,

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 :

\int_a^b f(x) g(x) \, dx = f(c) \int_a^b  g(x) \, dx.

Proof that there is some ''c'' in 'a'', ''b''

Suppose ''f'' : 'a'', ''b''→ R is continuous and ''g'' is a nonnegative integrable function on 'a'', ''b'' By the

extreme value theorem In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> ...

, there exists ''m'' and ''M'' such that for each ''x'' in 'a'', ''b''

m\leq f(x) \leq M

and

f,b =

, M The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline of ...

/math>. Since ''g'' is nonnegative, :

m \int_a^b g(x) \, dx \leq \int^b_a f(x) g(x) \, dx \leq M \int_a^b g(x) \, dx.

Now let :

I = \int_a^b g(x) \, dx.

I = 0

, we're done since :

0 \leq \int_a^b f(x) g(x)\, dx \leq 0

means :

\int_a^b f(x)g(x)\, dx=0,

so for any ''c'' in (''a'', ''b''), :

\int_a^b f(x)g(x)\, dx = f(c) I = 0.

If ''I'' ≠ 0, then :

m \leq \frac \int_a^b f(x)g(x)\,dx \leq M.

By the

intermediate value theorem In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two im ...

, ''f'' attains every value of the interval 'm'', ''M'' so for some ''c'' in 'a'', ''b'' :

f(c) = \frac1I\int^b_a f(x) g(x) \, dx,

that is, :

\int_a^b f(x) g(x) \, dx = f(c) \int_a^b  g(x) \, dx.

Finally, if ''g'' is negative on 'a'', ''b'' then :

M \int_a^b g(x) \, dx \leq \int^b_a f(x)g(x) \, dx \leq m \int_a^b g(x) \, dx,

and we still get the same result as above. QED

Second mean value theorem for definite integrals

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 In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...

function and φ : 'a'', ''b''→ R is an integrable function, then there exists a number ''x'' in (''a'', ''b''] such that ::

\int_a^b G(t)\varphi(t)\,dt = G(a^+) \int_a^x \varphi(t)\,dt.

Here

G(a^+)

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 monotone function, 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 ::

\int_a^b G(t)\varphi(t)\,dt = G(a^+) \int_a^x \varphi(t)\,dt + G(b^-) \int_x^b \varphi(t)\,dt.

Mean value theorem for integration fails for vector-valued functions

If the function

G

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

G

is also multi-dimensional. For example, consider the following 2-dimensional function defined on an

n

-dimensional cube: :

n \to \R^2 \\ G(x_1, \dots, x_n) = \left(\sin(x_1 + \cdots + x_n), \cos(x_1 + \cdots + x_n) \right) \end

Then, by symmetry it is easy to see that the mean value of

G

over its domain is (0,0): :

\int_ G(x_1,\dots,x_n) dx_1 \cdots dx_n = (0,0)

However, there is no point in which

G=(0,0)

, because

, G, =1

everywhere.

A probabilistic analogue of the mean value theorem

Let ''X'' and ''Y'' be non-negative

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

s such that E 'X''< E 'Y''< ∞ and

X \leq_ Y

(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 In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...

f_Z(x)=\,, \qquad x\geqslant 0.

Let ''g'' be a

measurable In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...

and

differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...

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 :

(Y) A thumb signal, usually described as a thumbs-up or thumbs-down, is a common hand gesture achieved by a closed fist held with the thumb extended upward or downward in approval or disapproval, respectively. These gestures have become metaphors ...

(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...

'(Z) The apostrophe ( or ) is a punctuation mark, and sometimes a diacritical mark, in languages that use the Latin alphabet and some other alphabets. In English, the apostrophe is used for two basic purposes: * The marking of the omission of one o ...

, Y)-(X)

Mean value theorem in complex variables

As noted above, the theorem does not hold for differentiable complex-valued functions. Instead, a generalization of the theorem is stated such:1 J.-Cl. Evard, F. Jafari, A Complex Rolle’s Theorem, American Mathematical Monthly, Vol. 99, Issue 9, (Nov. 1992), pp. 858-861. Let ''f'' : Ω → C be a

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...

on the open convex set Ω, and let ''a'' and ''b'' be distinct points in Ω. Then there exist points ''u'', ''v'' on the interior of the line segment from ''a'' to ''b'' such that :

\operatorname(f'(u)) = \operatorname\left ( \frac \right),

\operatorname(f'(v)) = \operatorname\left ( \frac \right).

Where Re() is the real part and Im() is the imaginary part of a complex-valued function. See also: Voorhoeve index.

Notes

References

External links

*
PlanetMath: Mean-Value Theorem
* *
"Mean Value Theorem: Intuition behind the Mean Value Theorem"
at the

Khan Academy Khan Academy is an American non-profit educational organization created in 2008 by Sal Khan. Its goal is creating a set of online tools that help educate students. The organization produces short lessons in the form of videos. Its website also i ...

{{Authority control Augustin-Louis Cauchy Articles containing proofs Theorems in calculus Theorems in real analysis