View source for Mean value theorem

{{For|the theorem in harmonic function theory|Harmonic function#The mean value property}}
{{Calculus}}
[[Image:Mvt2.svg|thumb|260 px|right|For any function that is continuous on <math>[a,b]</math> and differentiable on <math>(a,b)</math> there exists some <math>c</math> in the interval <math>(a,b)</math> such that the secant joining the endpoints of the interval <math>[a,b]</math> is parallel to the tangent at <math>c</math> .]]

In [[mathematics]], the '''mean value theorem''' states, roughly, that for a given planar [[arc (geometry)|arc]] between two endpoints, there is at least one point at which the [[tangent]] to the arc is parallel to the [[secant line|secant]] through its endpoints.

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, if <math>f</math> is a [[continuous function]] on the [[Interval (mathematics)#Terminology|closed interval]] <math>[a,b]</math>, and differentiable on the [[Interval (mathematics)#Terminology|open interval]] <math>(a,b)</math>, then there exists a point <math>c</math> in <math>(a,b)</math> such that:<ref name=mathworld>{{cite web|title=Mean-Value Theorem|last=Weisstein|first=Eric|url=http://mathworld.wolfram.com/Mean-ValueTheorem.html|work=[[MathWorld]]|publisher=[[Wolfram Research]]|accessdate=24 March 2011}}</ref>

:<math>f'(c)=\frac{f(b)-f(a)}{b-a}.</math>

It is one of the most important results in [[real analysis]].

== History ==

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 [[Bhāskara II]].<ref>J. J. O'Connor and E. F. Robertson (2000). [http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Paramesvara.html Paramesvara], ''[[MacTutor History of Mathematics archive]]''.</ref> 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.<ref>{{cite web|author=Ádám Besenyei|title=Historical development of the mean value theorem| url=http://abesenyei.web.elte.hu/publications/meanvalue.pdf}}</ref>

== Formal statement ==
[[File:Mittelwertsatz3.svg|thumb|The function <math>f</math> attains the slope of the secant between <math>a</math> and <math>b</math> as the derivative at the point <math>\xi\in(a,b)</math>.]]
[[File:Mittelwertsatz6.svg|thumb|It is also possible that there are multiple tangents parallel to the secant.]]
Let <math>f:[a,b]\to\R</math> be a [[continuous function]] on the closed [[interval (mathematics)|interval]] <math>[a,b]</math> , and [[differentiable function|differentiable]] on the open interval <math>(a,b)</math>, where <math>a<b</math>. Then there exists some <math>c</math> in <math>(a,b)</math> such that

:<math>f'(c)=\frac{f(b)-f(a)}{b-a}.</math>

The mean value theorem is a generalization of [[Rolle's theorem]], which assumes <math>f(a)=f(b)</math>, 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 <math>f:[a,b]\to\R</math> is [[continuous function|continuous]] on <math>[a,b]</math> , and that for every <math>x</math> in <math>(a,b)</math> the [[limit of a function|limit]]

:<math>\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}</math>

exists as a finite number or equals <math>\infty</math> or <math>-\infty</math> . If finite, that limit equals <math>f'(x)</math> . An example where this version of the theorem applies is given by the real-valued [[cube root]] function mapping <math>x \to x^\frac13</math> , 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 <math>f(x)=e^{xi}</math> for all real <math>x</math> . Then
:<math>f(2\pi)-f(0)=0=0(2\pi-0)</math>
while <math>f'(x)\ne 0</math> for any real <math>x</math> .

These formal statements are also known as Lagrange's Mean Value Theorem.<ref name=":0">{{Cite book|url=https://books.google.com/?id=e27uJruMCBUC&q=extended+mean+theorem#v=onepage&q=mean&f=false|title=Kirshna's Real Analysis: (General)|publisher=Krishna Prakashan Media|language=en}}</ref>

==Proof==
The expression <math>\frac{f(b)-f(a)}{b-a}</math> gives the [[slope]] of the line joining the points <math>(a,f(a))</math> and <math>(b,f(b))</math> , which is a [[Chord (geometry)|chord]] of the graph of <math>f</math> , while <math>f'(x)</math> gives the slope of the tangent to the curve at the point <math>(x,f(x))</math> . 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 <math>g(x)=f(x)-rx</math> , where <math>r</math> is a constant. Since <math>f</math> is continuous on <math>[a,b]</math> and differentiable on <math>(a,b)</math> , the same is true for <math>g</math> . We now want to choose <math>r</math> so that <math>g</math> satisfies the conditions of [[Rolle's theorem]]. Namely

:<math>\begin{align}g(a)=g(b)&\iff f(a)-ra=f(b)-rb\\ &\iff r(b-a)=f(b)-f(a) \\&\iff r=\frac{f(b)-f(a)}{b-a}\cdot\end{align}</math>

By [[Rolle's theorem]], since <math>g</math> is differentiable and <math>g(a)=g(b)</math> , there is some <math>c</math> in <math>(a,b)</math> for which <math>g'(c)=0</math> , and it follows from the equality <math>g(x)=f(x)-rx</math> that,

:<math>\begin{align}&g'(x) = f'(x) -r \\& g'(c) = 0\\&g'(c) = f'(c) - r = 0 \\&\Rightarrow f'(c) = r = \frac{f(b)-f(a)}{b-a} \end{align}
</math>

==A simple application==
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 (topology)|interior point]] of the interval ''I'' exists and is zero, then ''f'' is [[constant function|constant]] in the interior.

'''Proof:''' Assume the derivative of ''f'' at every [[interior (topology)|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

:<math>0=f'(c)=\frac{f(b)-f(a)}{b-a}.</math>

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 ''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 [[Derivative#Continuity and differentiability|continuity and differentiability]] of the article [[derivative]].)
* The differentiability of ''f'' can be relaxed to [[one-sided derivatives|one-sided differentiability]], a proof given in the article on [[semi-differentiability]].

==Cauchy's mean value theorem==<!-- This section is linked from [[Taylor's theorem]] -->
'''Cauchy's mean value theorem''', also known as the '''extended mean value theorem''',<ref>{{Cite web|url=http://mathworld.wolfram.com/ExtendedMean-ValueTheorem.html|title=Extended Mean-Value Theorem|last=W.|first=Weisstein, Eric|website=mathworld.wolfram.com|language=en|access-date=2018-10-08}}</ref> 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<ref name=":0" />
[[Image:Cauchy.svg|right|thumb|260px|Geometrical meaning of Cauchy's theorem]]

:<math>(f(b)-f(a))g'(c)=(g(b)-g(a))f'(c).</math>

Of course, if {{nowrap|''g''(''a'') ≠ ''g''(''b'')}} and if {{nowrap|''g′''(''c'') ≠ 0}}, this is equivalent to:

:<math>\frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}.</math>

Geometrically, this means that there is some [[tangent]] to the graph of the [[curve]]<ref>{{Cite news|url=https://www.math24.net/cauchys-mean-value-theorem/|title=Cauchy's Mean Value Theorem|work=Math24|access-date=2018-10-08|language=en-US}}</ref>

:<math>\begin{cases}[a,b] \to \mathbf{R}^2\\t\mapsto (f(t),g(t))\end{cases}</math>

which is [[Parallel (geometry)|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 {{nowrap|''f′''(''c'') {{=}} ''g′''(''c'') {{=}} 0}}, in other words a value for which the mentioned curve is [[Stationary point|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

:<math>t\mapsto(t^3,1-t^2),</math>

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 (singularity)|cusp]]) at {{nowrap|''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 {{nowrap|''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'') ≠ ''g''(''b''). Define ''h''(''x'') = ''f''(''x'') − ''rg''(''x''), where ''r'' is fixed in such a way that ''h''(''a'') = ''h''(''b''), namely
:: <math>\begin{align}h(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{f(b)-f(a)}{g(b)-g(a)}.\end{align}</math>
: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:
::<math>0=h'(c)=f'(c)-rg'(c) = f'(c)- \left (\frac{f(b)-f(a)}{g(b)-g(a)} \right ) g'(c).</math>
:Therefore:
::<math>f'(c)= \frac{f(b)-f(a)}{g(b)-g(a)} g'(c),</math>
:which implies the result.<ref name=":0" />

* 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==
Assume that <math>f, g,</math> and <math>h</math> are differentiable functions on <math>(a,b)</math> that are continuous on <math>[a,b]</math>. Define

:<math> D(x)=\left |\begin{array}{ccc}f(x) & g(x)& h(x)\\ f(a) & g(a) & h(a)\\ f(b) & g(b)& h(b)\end{array}\right|</math>

There exists <math>c\in(a,b)</math> such that <math> D'(c)=0</math>.

Notice that
:<math> D'(x)=\left |\begin{array}{ccc}f'(x) & g'(x)& h'(x)\\ f(a) & g(a) & h(a)\\ f(b) & g(b)& h(b)\end{array}\right|</math>
and if we place <math>h(x)=1</math>, we get Cauchy's mean value theorem. If we place <math>h(x)=1</math> and <math>g(x)=x</math> we get '''Lagrange's mean value theorem'''.

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

== 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 <math>G</math> be an open convex subset of <math>\R^n</math> , and let <math>f:G\to\R</math> be a differentiable function. Fix points <math>x,y\in G</math> , and define <math>g(t)=f\Big((1-t)x+ty\Big)</math> . Since <math>g</math> is a differentiable function in one variable, the mean value theorem gives:

:<math>g(1)-g(0)=g'(c)</math>

for some <math>c</math> between 0 and 1. But since <math>g(1)=f(y)</math> and <math>g(0)=f(x)</math> , computing <math>g'(c)</math> explicitly we have:

:<math>f(y)-f(x)=\nabla f\Big((1-c)x+cy\Big)\cdot (y-x)</math>

where <math>\nabla</math> denotes a [[gradient]] and <math>\cdot</math> a [[dot product]]. Note that this is an exact analog of the theorem in one variable (in the case <math>n=1</math> this ''is'' the theorem in one variable). By the [[Cauchy–Schwarz inequality]], the equation gives the estimate:

:<math>\Bigg|f(y)-f(x)\Bigg|\le\Bigg|\nabla f\Big((1-c)x+cy\Big)\Bigg|\ \Big|y - x\Big|.</math>

In particular, when the partial derivatives of <math>f</math> are bounded, <math>f</math> is [[Lipschitz continuity|Lipschitz continuous]] (and therefore [[Uniform continuity|uniformly continuous]]). Note that <math>f</math> is not assumed to be continuously differentiable or continuous on the closure of <math>G</math> . However, in order to use the chain rule to compute <math> g' </math>, we really do need to know that <math>f</math> is differentiable on <math>G</math>; the existence of the <math> x </math> and <math> y </math> partial derivatives by itself is not sufficient for the theorem to be true {{clarify|date=March 2018| reason=Sufficient for what to be true?}}.

As an application of the above, we prove that <math>f</math> is constant if <math>G</math> is open and connected and every partial derivative of <math>f</math> is 0. Pick some point <math>x_0\in G</math> , and let <math>g(x)=f(x)-f(x_0)</math> . We want to show <math>g(x)=0</math> for every <math>x\in G</math> . For that, let <math>E=\{x\in G:g(x)=0\}</math> . Then ''E'' is closed and nonempty. It is open too: for every <math>x\in E</math> ,

:<math>\Big|g(y)\Big|=\Bigg|g(y)-g(x)\Bigg|\le (0)\Big|y-x\Big|=0</math>

for every <math>y</math> in some neighborhood of <math>x</math> . (Here, it is crucial that <math>x</math> and <math>y</math> are sufficiently close to each other.) Since <math>G</math> is connected, we conclude <math>E=G</math> .

The above arguments are made in a coordinate-free manner; hence, they generalize to the case when <math>G</math> 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,'' [[Walter Rudin|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:<ref>{{Cite book|url=https://archive.org/details/1979RudinW|title=Principles of Mathematical Analysis (3rd ed.)|last=Rudin|first=Walter|publisher=McGraw-Hill|year=1976|isbn=978-0-07-054235-8|location=New York|pages=113|quote=|via=}}</ref>

'''Theorem.'''  ''For a continuous vector-valued function <math>\mathbf{f}:[a,b]\to\mathbb{R}^k</math> differentiable on <math>(a,b)</math>, there exists <math>x\in(a,b)</math> such that <math>|\mathbf{f}'(x)|\geq\frac{1}{b-a}|\mathbf{f}(b)-\mathbf{f}(a)| </math>.''

[[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'' → '''R'''<sup>''m''</sup> is a differentiable function (where ''U'' ⊂ '''R'''<sup>''n''</sup> is open) and if ''x'' + ''th'', ''x, h'' ∈ '''R'''<sup>''n''</sup>, ''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 ''f<sub>i</sub>'' (''i'' = 1, ..., ''m'') of ''f'' (in the above notation set ''y'' = ''x'' + ''h''). In doing so one finds points ''x'' + ''t<sub>i</sub>h'' on the line segment satisfying

:<math>f_i(x+h) - f_i(x) = \nabla f_i (x + t_ih) \cdot h.</math>

But generally there will not be a ''single'' point ''x'' + ''t*h'' on the line segment satisfying

:<math>f_i(x+h) - f_i(x) = \nabla f_i (x + t^* h) \cdot h.</math>

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

:<math>\begin{cases} f : [0, 2 \pi] \to \mathbf{R}^2 \\ f(x) = (\cos(x), \sin(x)) \end{cases}</math>

Then <math>f(2\pi) - f(0) = \mathbf{0} \in \mathbf{R}^2</math>, but <math>f_1'(x)=-\sin (x)</math> and <math>f_2'(x)=\cos (x)</math> are never simultaneously zero as <math>x</math> ranges over <math>\left[0, 2 \pi\right]</math>.

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

:<math>f(x+h)-f(x) = f'(x+t^*h)\cdot h.</math>

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

:<math> f(x+h)-f(x) = \int_x^{x+h} f'(u) \, du = \left (\int_0^1 f'(x+th)\,dt\right)\cdot h.</math>

Thus, the value ''f′''(''x'' + ''t*h'') at the particular point ''t*'' has been replaced by the mean value

:<math>\int_0^1 f'(x+th)\,dt.</math>

This last version can be generalized to vector valued functions:

:'''Lemma 1.''' Let ''U'' ⊂ '''R'''<sup>''n''</sup> be open, ''f'' : ''U'' → '''R'''<sup>''m''</sup> continuously differentiable, and ''x'' ∈ ''U'', ''h'' ∈ '''R'''<sup>''n''</sup> vectors such that the line segment ''x'' + ''th'',  0 ≤ ''t'' ≤ 1 remains in ''U''. Then we have:
::<math>f(x+h)-f(x) = \left (\int_0^1 Df(x+th)\,dt\right)\cdot h,</math>
:where ''Df'' denotes the [[Jacobian matrix]] of ''f'' and the integral of a matrix is to be understood componentwise.

'''Proof.''' Let ''f''<sub>1</sub>, ..., ''f<sub>m</sub>'' denote the components of ''f'' and define:

:<math>\begin{cases} g_i : [0,1] \to \mathbf{R} \\ g_i(t) = f_i (x +th) \end{cases} </math>

Then we have

: <math>
\begin{align}
& 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_{j=1}^n \frac{\partial f_i}{\partial x_j} (x+th)h_j\right)\,dt = \sum_{j=1}^n \left (\int_0^1 \frac{\partial f_i}{\partial x_j}(x+th)\,dt\right)h_j.
\end{align}
</math>

The claim follows since ''Df'' is the matrix consisting of the components <math>\tfrac{\partial f_i}{\partial x_j}.</math>

:'''Lemma 2.''' Let ''v'' : [''a'', ''b''] → '''R'''<sup>''m''</sup> be a continuous function defined on the interval [''a'', ''b''] ⊂ '''R'''. Then we have
::<math>\left \|\int_a^b v(t)\,dt\right\|\leqslant \int_a^b \|v(t)\|\,dt.</math>

'''Proof.''' Let ''u'' in '''R'''<sup>''m''</sup> denote the value of the integral
:<math>u:=\int_a^b v(t)\,dt.</math>
Now we have (using the [[Cauchy–Schwarz inequality]]):
:<math>\|u\|^2 = \langle u,u \rangle =\left \langle \int_a^b v(t) \, dt, u \right\rangle = \int_a^b \langle v(t),u \rangle \,dt \leqslant \int_a^b \| v(t) \|\cdot \|u \|\,dt = \|u\| \int_a^b \|v(t)\|\,dt</math>
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
::<math>\|f(x+h)-f(x)\| \leqslant M\|h\|.</math>

'''Proof.''' From Lemma 1 and 2  it follows that
:<math>\|f(x+h)-f(x)\|=\left \|\int_0^1 (Df(x+th)\cdot h)\,dt\right\|  \leqslant \int_0^1 \|Df(x+th)\| \cdot \|h\|\, dt \leqslant M\| h\|.</math>

== Mean value theorems for definite integrals ==

===First mean value theorem for definite integrals===
[[File:积分中值定理.jpg|thumb|right|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''<ref>{{cite web|url=http://www.mathwords.com/m/mean_value_theorem_integrals.htm|title=Mathwords: Mean Value Theorem for Integrals|author=|date=|website=www.mathwords.com}}</ref>]]
Let ''f'' : [''a'', ''b''] → '''R''' be a continuous function. Then there exists ''c'' in (''a'', ''b'') such that

:<math>\int_a^b f(x) \, dx = f(c)(b - a).</math>

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

:<math>\frac{1}{b-a} \int_a^b f(x) \, dx,</math>

we can interpret the conclusion as ''f'' achieves its mean value at some ''c'' in (''a'', ''b'').<ref name="Comenetz2002">{{cite book|author=Michael Comenetz|title=Calculus: The Elements| year=2002| publisher=World Scientific|isbn=978-981-02-4904-5|page=159}}</ref>

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

:<math>\int_a^b f(x) g(x) \, dx = f(c) \int_a^b  g(x) \, dx.</math>

===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''], <math>m\leqslant  f(x) \leqslant  M </math> and <math>f[a,b] = [m, M]</math>. Since ''g'' is nonnegative,

:<math>m \int_a^b g(x) \, dx \leqslant  \int^b_a f(x)g(x) \, dx \leqslant  M \int_a^b g(x) \, dx.</math>

Now let

:<math>I = \int_a^b g(x) \, dx.</math>

If <math>I = 0</math>, we're done since

:<math>0 \leqslant  \int_a^b f(x) g(x)\, dx \leqslant  0</math>

means

:<math>\int_a^b f(x)g(x)\, dx=0,</math>

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

:<math>\int_a^b f(x)g(x)\, dx = f(c) I = 0.</math>

If ''I'' ≠ 0, then

:<math>m \leqslant  \frac1I \int_a^b f(x)g(x)\,dx \leqslant  M.</math>

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

:<math>f(c) = \frac1I\int^b_a f(x) g(x) \, dx,</math>

that is,

:<math>\int_a^b f(x) g(x) \, dx = f(c) \int_a^b  g(x) \, dx.</math>

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

:<math>M \int_a^b g(x) \, dx \leqslant  \int^b_a f(x)g(x) \, dx \leqslant  m \int_a^b g(x) \, dx,</math>

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 [[monotone function|monotonically decreasing]] function and φ : [''a'', ''b''] → '''R''' is an integrable function, then there exists a number ''x'' in (''a'', ''b''] such that
::<math> \int_a^b G(t)\varphi(t)\,dt = G(a^+) \int_a^x \varphi(t)\,dt. </math>

Here <math>G(a^+)</math> stands for <math>{\lim_{x\to a^+}G(x)}</math>, 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:<ref>{{cite journal |first=E. W. |last=Hobson |year=1909 |title=On the Second Mean-Value Theorem of the Integral Calculus |journal=[[Proceedings London Mathematical Society|Proc. London Math. Soc.]] |volume=S2–7 |issue=1 |pages=14–23 |mr=1575669 |doi=10.1112/plms/s2-7.1.14 |url=https://zenodo.org/record/1447800 }}</ref>

: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
::<math> \int_a^b G(t)\varphi(t)\,dt = G(a^+) \int_a^x \varphi(t)\,dt + G(b^-) \int_x^b \varphi(t)\,dt. </math>

=== Mean value theorem for integration fails for vector-valued functions ===
If the function <math>G</math> returns a multi-dimensional vector, then the MVT for integration is not true, even if the domain of <math>G</math> is also multi-dimensional.

For example, consider the following 2-dimensional function defined on an <math>n</math>-dimensional cube:

:<math>\begin{cases} G: [0,2\pi]^n \to \mathbb{R}^2 \\ G(x_1,\cdots,x_n)=\left (\sin(x_1+\cdots+x_n), \cos(x_1+\cdots+x_n)\right)\end{cases} </math>

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

:<math>\int_{[0,2\pi]^n} G(x_1,\cdots,x_n) dx_1 \cdots dx_n = (0,0)</math>

However, there is no point in which <math>G=(0,0)</math>, because <math>|G|=1</math> everywhere.

==A probabilistic analogue of the mean value theorem==
Let ''X'' and ''Y'' be non-negative [[random variable]]s such that E[''X''] < E[''Y''] < ∞ and <math>X\leqslant_{st} Y</math> (i.e. ''X'' is smaller than ''Y'' in the [[Stochastic ordering|usual stochastic order]]).  Then there exists an absolutely continuous non-negative random variable ''Z'' having [[probability density function]]

:<math> f_Z(x)={\Pr(Y>x)-\Pr(X>x)\over {\rm E}[Y]-{\rm E}[X]}\,, \qquad x\geqslant 0.</math>

Let ''g'' be a [[Measurable function|measurable]] and [[differentiable function]] such that E[''g''(''X'')], E[''g''(''Y'')] < ∞, and let its derivative ''g′''  be measurable and [[Riemann integral|Riemann-integrable]]  on the interval [''x'', ''y''] for all ''y'' ≥ ''x'' ≥ 0.  Then, E[''g′''(''Z'')] is finite and<ref>{{cite journal |first=A. |last=Di Crescenzo |year=1999 |title=A Probabilistic Analogue of the Mean Value Theorem and Its Applications to Reliability Theory |journal=[[J. Appl. Probab.]] |volume=36 |issue=3 |pages=706–719 |jstor=3215435 |doi=10.1239/jap/1032374628 }}</ref>

:<math> {\rm E}[g(Y)]-{\rm E}[g(X)]={\rm E}[g'(Z)]\,[{\rm E}(Y)-{\rm E}(X)].</math>

== Generalization in complex analysis ==
As noted above, the theorem does not hold for differentiable complex-valued functions. Instead, a generalization of the theorem is stated such:<ref name=PlanetMath>{{cite web|title=Complex Mean-Value Theorem|url=http://planetmath.org/ComplexMeanValueTheorem.html|work=[[PlanetMath]]|publisher=[[PlanetMath]]}}</ref>

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 ''L<sub>ab</sub>'' (the line segment from ''a'' to ''b'') such that
:<math>\mathrm{Re}(f'(u)) = \mathrm{Re}\left ( \frac{f(b)-f(a)}{b-a} \right),</math>
:<math>\mathrm{Im}(f'(v)) = \mathrm{Im}\left ( \frac{f(b)-f(a)}{b-a} \right).</math>

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

== See also ==
* [[Newmark-beta method]]
* [[Mean value theorem (divided differences)]]
* [[Racetrack principle]]
* [[Stolarsky mean]]

== Notes ==
{{reflist}}

== External links ==
* {{springerEOM|title=Cauchy theorem|id=p/c020990}}
* [http://planetmath.org/encyclopedia/MeanValueTheorem.html PlanetMath: Mean-Value Theorem]
* {{MathWorld|Mean-ValueTheorem|Mean value theorem}}
* {{MathWorld|CauchysMean-ValueTheorem|Cauchy's Mean-Value Theorem}}
* [http://www.khanacademy.org/video/mean-value-theorem "Mean Value Theorem: Intuition behind the Mean Value Theorem"] at the [[Khan Academy]]

{{DEFAULTSORT:Mean Value Theorem}}
[[Category:Augustin-Louis Cauchy]]
[[Category:Articles containing proofs]]
[[Category:Theorems in calculus]]
[[Category:Theorems in real analysis]]