mean value theorem#Cauchy's mean value theorem

A special case of this theorem for inverse interpolation of the sine was first described by Parameshvara (1380–1460), from the Kerala School of Astronomy and Mathematics in India, in his commentaries on Govindasvāmi and Bhāskara II.J. J. O'Connor and E. F. Robertson (2000). [https://mathshistory.st-andrews.ac.uk/Biographies/Paramesvara/ Paramesvara], MacTutor History of Mathematics archive. 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.{{cite web|author=Ádám Besenyei|title=Historical development of the mean value theorem| url=http://abesenyei.web.elte.hu/publications/meanvalue.pdf}} Many variations of this theorem have been proved since then.{{Cite journal|last=Lozada-Cruz|first=German|date=2020-10-02|title=Some variants of Cauchy's mean value theorem|url=https://www.tandfonline.com/doi/full/10.1080/0020739X.2019.1703150|journal=International Journal of Mathematical Education in Science and Technology|language=en|volume=51|issue=7|pages=1155–1163|doi=10.1080/0020739X.2019.1703150|bibcode=2020IJMES..51.1155L|s2cid=213335491|issn=0020-739X}}{{Cite book|last=Sahoo, Prasanna.|url=https://www.worldcat.org/oclc/40951137|title=Mean value theorems and functional equations|date=1998|publisher=World Scientific|others=Riedel, T. (Thomas), 1962-|isbn=981-02-3544-5|location=Singapore|oclc=40951137}}

File:Mittelwertsatz3.svg

File:Mittelwertsatz6.svg

Let $f:[a,b]\backslash to\backslash R$ be a continuous function on the closed interval {{nowrap|$[a,b]$,}} and differentiable on the open interval {{nowrap|$(a,b)$,}} where {{nowrap|.}} Then there exists some $c$ in $(a,b)$ such that:{{sfn|Rudin|1976|p=108}}

:$f\text{'}(c)=\backslash frac\{f(b)-f(a)\}\{b-a\}.$

The mean value theorem is a generalization of Rolle's theorem, 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 $f:[a,b]\backslash to\backslash R$ is continuous on $[a,b]$, and that for every $x$ in $(a,b)$ the limit

:$\backslash lim\_\{h\backslash to\; 0\}\backslash frac\{f(x+h)-f(x)\}\{h\}$

exists as a finite number or equals $\backslash infty$ or $-\backslash infty$. If finite, that limit equals $f\text{'}(x)$. An example where this version of the theorem applies is given by the real-valued cube root function mapping $x\; \backslash mapsto\; x^\{1/3\}$, whose derivative tends to infinity at the origin.

The expression $\backslash frac\{f(b)-f(a)\}\{b-a\}$ 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 $f\text{'}(x)$ 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 $[a,b]$ 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

:$\backslash begin\{align\}$

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} .

\end{align}

By Rolle's theorem, since $g$ is differentiable and $g(a)=g(b)$, there is some $c$ in $(a,b)$ for which $g\text{'}(c)=0$ , and it follows from the equality $g(x)=f(x)-rx$ that,

:$\backslash begin\{align\}$

&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}

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 on $I$.

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

:$0=f\text{'}(c)=\backslash frac\{f(b)-f(a)\}\{b-a\}.$

This implies that {{nowrap|1=$f(a)\; =\; f(b)$}}. 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 continuity and differentiability of the article derivative.)
• The differentiability of $f$ can be relaxed to one-sided differentiability, a proof is given in the article on semi-differentiability.

Theorem 2: If $f\text{'}(x)\; =\; g\text{'}(x)$ 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 $F(x)\; =\; f(x)\; -\; g(x)$, then $F\text{'}(x)=f\text{'}(x)-g\text{'}(x)=0$ on the interval $(a,\; b)$, so the above theorem 1 tells that $F(x)\; =\; f(x)\; -\; g(x)$ 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, also known as the extended mean value theorem, is a generalization of the mean value theorem.{{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}}{{sfn|Rudin|1976|pp=107-108}} It states: if the 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 $c\; \backslash in\; (a,b)$, such that

Image:Cauchy.svg

:$(f(b)-f(a))g\text{'}(c)=(g(b)-g(a))f\text{'}(c).$

Of course, if $g(a)\; \backslash neq\; g(b)$ and $g\text{'}(c)\; \backslash neq\; 0$, this is equivalent to:

:$\backslash frac\{f\text{'}(c)\}\{g\text{'}(c)\}=\backslash frac\{f(b)-f(a)\}\{g(b)-g(a)\}.$

Geometrically, this means that there is some tangent to the graph of the curve{{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}}

:$\backslash begin\{cases\}[a,b]\; \backslash to\; \backslash R^2\backslash \backslash t\backslash mapsto\; (f(t),g(t))\backslash end\{cases\}$

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 $f\text{'}(c)\; =\; g\text{'}(c)\; =\; 0$, 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

:$t\; \backslash mapsto\; \backslash left(t^3,1-t^2\backslash right),$

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 $g(t)\; =\; t$.

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

Define $h(x)\; =\; (g(b)-g(a))f(x)\; -\; (f(b)-f(a))g(x)$, then we easily see $h(a)=h(b)=f(a)g(b)-f(b)g(a)$.

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 $h\text{'}(c)\; =\; 0$. Now using the definition of $h$ we have:

:$0=h\text{'}(c)=(g(b)-g(a))f\text{'}(c)-(f(b)-f(a))g\text{'}(c)$

The result easily follows.

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 $\backslash R^n$, and let $f:G\backslash to\backslash R$ be a differentiable function. Fix points $x,y\backslash in\; G$ such that the line segment between $x,\; y$ lies in $G$, and define $g(t)=f\backslash big((1-t)x+ty\backslash big)$. Since $g$ is a differentiable function in one variable, the mean value theorem gives:

:$g(1)-g(0)=g\text{'}(c)$

for some $c$ between 0 and 1. But since $g(1)=f(y)$ and $g(0)=f(x)$, computing $g\text{'}(c)$ explicitly we have:

:$f(y)-f(x)=\backslash nabla\; f\backslash big((1-c)x+cy\backslash big)\backslash cdot\; (y-x)$

where $\backslash nabla$ denotes a gradient and $\backslash cdot$ a dot product. 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 equation gives the estimate:

:$\backslash Bigl|f(y)-f(x)\backslash Bigr|\; \backslash le\; \backslash Bigl|\backslash nabla\; f\backslash big((1-c)x+cy\backslash big)\backslash Bigr|\backslash \; \backslash Bigl|y\; -\; x\backslash Bigr|.$

In particular, when $G$ is convex and the partial derivatives of $f$ are bounded, $f$ is Lipschitz continuous (and therefore uniformly continuous).

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 $x\_0\backslash in\; G$, and let $g(x)=f(x)-f(x\_0)$. We want to show $g(x)=0$ for every $x\backslash in\; G$. For that, let $E=\backslash \{x\backslash in\; G:g(x)=0\backslash \}$. Then $E$ is closed in $G$ and nonempty. It is open too: for every $x\backslash in\; E$ ,

:$\backslash Big|g(y)\backslash Big|=\backslash Big|g(y)-g(x)\backslash Big|\backslash le\; (0)\backslash Big|y-x\backslash Big|=0$

for every $y$ in open ball centered at $x$ and contained in $G$. 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.

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:{{sfn|Rudin|1976|p=113}}

{{math theorem|For a continuous vector-valued function $\backslash mathbf\{f\}:[a,b]\backslash to\backslash mathbb\{R\}^k$ differentiable on $(a,b)$, there exists a number $c\backslash in(a,b)$ such that

:$|\backslash mathbf\{f\}(b)-\backslash mathbf\{f\}(a)|\; \backslash le\; (b-a)\backslash left|\backslash mathbf\{f\}\text{'}(c)\backslash right|$.}}

{{math proof| Take $\backslash varphi(t)\; =\; (\backslash textbf\{f\}(b)\; -\; \backslash textbf\{f\}(a))\; \backslash cdot\; \backslash textbf\{f\}(t)$. Then $\backslash varphi$ is real-valued and thus, by the mean value theorem,

:$\backslash varphi(b)\; -\; \backslash varphi(a)\; =\; \backslash varphi\text{'}(c)(b-a)$

for some $c\; \backslash in\; (a,\; b)$. Now,

$$\backslash varphi(b)\; -\; \backslash varphi(a)\; =\; |\backslash textbf\{f\}(b)\; -\; \backslash textbf\{f\}(a)|^2$$

and

$$\backslash varphi\text{'}(c)\; =\; (\backslash textbf\{f\}(b)\; -\; \backslash textbf\{f\}(a))\; \backslash cdot\; \backslash textbf\{f\}\text{'}(c).$$

Hence, using the Cauchy–Schwarz inequality, from the above equation, we get:

:$|\backslash textbf\{f\}(b)\; -\; \backslash textbf\{f\}(a)|^2\; \backslash le\; |\backslash textbf\{f\}(b)\; -\; \backslash textbf\{f\}(a)|\; |\backslash textbf\{f\}\text{'}(c)\; |(b-a).$

If $\backslash textbf\{f\}(b)\; =\; \backslash textbf\{f\}(a)$, the theorem holds trivially. Otherwise, dividing both sides by $|\backslash textbf\{f\}(b)\; -\; \backslash textbf\{f\}(a)|$ yields the theorem.}}

{{see also|Calculus on Euclidean space#Derivative of a map and chain rule}}

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 reason why there is no analog of mean value equality is the following: If {{math|f : U → R^m}} is a differentiable function (where {{math|U ⊂ Rⁿ}} is open) and if {{math|x + th}}, {{math|x, h ∈ Rⁿ, t ∈ [0, 1]}} is the line segment in question (lying inside {{mvar|U}}), then one can apply the above parametrization procedure to each of the component functions {{math|1=f_i (i = 1, …, m)}} of f (in the above notation set {{math|1=y = x + h}}). In doing so one finds points {{math|x + t_ih}} on the line segment satisfying

:$f\_i(x+h)\; -\; f\_i(x)\; =\; \backslash nabla\; f\_i\; (x\; +\; t\_ih)\; \backslash cdot\; h.$

But generally there will not be a single point {{math|x + t*h}} on the line segment satisfying

:$f\_i(x+h)\; -\; f\_i(x)\; =\; \backslash nabla\; f\_i\; (x\; +\; t^*\; h)\; \backslash cdot\; h.$

for all {{mvar|i}} simultaneously. For example, define:

:$\backslash begin\{cases\}$

f : [0, 2 \pi] \to \R^2 \\

f(x) = (\cos(x), \sin(x))

\end{cases}

Then $f(2\backslash pi)\; -\; f(0)\; =\; \backslash mathbf\{0\}\; \backslash in\; \backslash R^2$, but $f\_1\text{'}(x)\; =\; -\backslash sin\; (x)$ and $f\_2\text{'}(x)\; =\; \backslash cos\; (x)$ are never simultaneously zero as $x$ ranges over $\backslash left[0,\; 2\; \backslash pi\backslash right]$.

The above theorem implies the following:

{{math_theorem|name=Mean value inequality{{harvnb|Hörmander|2015|loc=Theorem 1.1.1. and remark following it.}} |math_statement=For a continuous function $\backslash textbf\{f\}\; :\; [a,\; b]\; \backslash to\; \backslash mathbb\{R\}^k$, if $\backslash textbf\{f\}$ is differentiable on $(a,\; b)$, then

:$|\backslash textbf\{f\}(b)\; -\; \backslash textbf\{f\}(a)|\; \backslash le\; (b-a)\backslash sup\_\{(a,\; b)\}\; |\backslash textbf\{f\}\text{'}|$.}}

In fact, the above statement suffices for many applications and can be proved directly as follows. (We shall write $f$ for $\backslash textbf\{f\}$ for readability.)

{{math proof|First assume $f$ is differentiable at $a$ too. If $f\text{'}$ is unbounded on $(a,\; b)$, there is nothing to prove. Thus, assume . Let $M\; >\; \backslash sup\_\{(a,\; b)\}\; |f\text{'}|$ be some real number. Let

$$E\; =\; \backslash \{\; 0\; \backslash le\; t\; \backslash le\; 1\; \backslash mid\; |f(a\; +\; t(b-a))\; -\; f(a)|\; \backslash le\; Mt(b-a)\; \backslash \}.$$

We want to show $1\; \backslash in\; E$. 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 $1\; \backslash in\; E$ and we are done. Thus suppose otherwise. For $1\; >\; t\; >\; s$,

:$\backslash begin\{align\}$

&|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{align}

Let $\backslash epsilon\; >\; 0$ be such that $M\; -\; \backslash epsilon\; >\; \backslash sup\_\{(a,\; b)\}\; |f\text{'}|$. 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 $\backslash le\; \backslash epsilon\; (t-s)(b-a)$. The second term is $\backslash le\; (M\; -\; \backslash epsilon)\; (t-s)(b-a)$. The third term is $\backslash le\; Ms(b-a)$. Hence, summing the estimates up, we get: $|f(a\; +\; t(b-a))\; -\; f(a)|\; \backslash le\; tM|b-a|$, a contradiction to the maximality of $s$. Hence, $1\; =\; s\; \backslash in\; M$ and that means:

:$|f(b)\; -\; f(a)|\; \backslash le\; M(b-a).$

Since $M$ is arbitrary, this then implies the assertion. Finally, if $f$ is not differentiable at $a$, let $a\text{'}\; \backslash in\; (a,\; b)$ and apply the first case to $f$ restricted on $[a\text{'},\; b]$, giving us:

:$|f(b)\; -\; f(a\text{'})|\; \backslash le\; (b-a\text{'})\backslash sup\_\{(a,\; b)\}\; |f\text{'}|$

since $(a\text{'},\; b)\; \backslash subset\; (a,\; b)$. Letting $a\text{'}\; \backslash to\; a$ finishes the proof.}}

All conditions for the mean value theorem are necessary:

1. $\backslash boldsymbol\{f(x)\}$ is differentiable on $\backslash boldsymbol\{(a,b)\}$
2. $\backslash boldsymbol\{f(x)\}$ is continuous on $\backslash boldsymbol\{[a,b]\}$
3. $\backslash boldsymbol\{f(x)\}$ is real-valued

When one of the above conditions is not satisfied, the mean value theorem is not valid in general, and so it cannot be applied.

thumb

The necessity of the first condition can be seen by the counterexample where the function $f(x)=|x|$ on [-1,1] is not differentiable.

The necessity of the second condition can be seen by the counterexample where the function

satisfies criteria 1 since $f\text{'}(x)=0$ on $(0,1)$ but not criteria 2 since

$$\backslash frac\{f(1)-f(0)\}\{1-0\}=-1$$ and $-1\backslash neq\; 0\; =f\text{'}(x)$ for all $x\backslash in\; (0,1)$ so no such $c$ exists.

The theorem is false if a differentiable function is complex-valued instead of real-valued. For example, if $f(x)\; =\; e^\{xi\}$ for all real $x$, then

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

while $f\text{'}(x)\backslash ne\; 0$ for any real $x$.

File:积分中值定理.jpg

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

:$\backslash int\_a^b\; f(x)\; \backslash ,\; dx\; =\; f(c)(b\; -\; a).$

This follows at once from the fundamental theorem of calculus, together with the mean value theorem for derivatives. Since the mean value of f on [a, b] is defined as

:$\backslash frac\{1\}\{b-a\}\; \backslash int\_a^b\; f(x)\; \backslash ,\; dx,$

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

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

:$\backslash int\_a^b\; f(x)\; g(x)\; \backslash ,\; dx\; =\; f(c)\; \backslash int\_a^b\; g(x)\; \backslash ,\; dx.$

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

:If $G\; :\; [a,b]\backslash to\; \backslash mathbb\{R\}$ is a positive monotonically decreasing function and $\backslash varphi\; :\; [a,b]\backslash to\; \backslash mathbb\{R\}$ is an integrable function, then there exists a number x in (a, b] such that

::$\backslash int\_a^b\; G(t)\backslash varphi(t)\backslash ,dt\; =\; G(a^+)\; \backslash int\_a^x\; \backslash varphi(t)\backslash ,dt.$

Here $G(a^+)$ stands for $\{\backslash lim\_\{x\backslash to\; a^+\}G(x)\}$, 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:{{cite journal |first=E. W. |last=Hobson |year=1909 |title=On the Second Mean-Value Theorem of the Integral Calculus |journal=Proc. London Math. Soc. |volume=S2–7 |issue=1 |pages=14–23 |mr=1575669 |doi=10.1112/plms/s2-7.1.14 |bibcode=1909PLMS...27...14H |url=https://zenodo.org/record/1447800 }}

:If $G\; :\; [a,b]\backslash to\; \backslash mathbb\{R\}$ is a monotonic (not necessarily decreasing and positive) function and $\backslash varphi\; :\; [a,b]\backslash to\; \backslash mathbb\{R\}$ is an integrable function, then there exists a number x in (a, b) such that

::$\backslash int\_a^b\; G(t)\backslash varphi(t)\backslash ,dt\; =\; G(a^+)\; \backslash int\_a^x\; \backslash varphi(t)\backslash ,dt\; +\; G(b^-)\; \backslash int\_x^b\; \backslash varphi(t)\backslash ,dt.$

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:

:$\backslash begin\{cases\}$

G: [0,2\pi]^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{cases}

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

:$\backslash int\_\{[0,2\backslash pi]^n\}\; G(x\_1,\backslash dots,x\_n)\; dx\_1\; \backslash cdots\; dx\_n\; =\; (0,0)$

However, there is no point in which $G=(0,0)$, because $|G|=1$ everywhere.

Assume that $f,\; g,$ and $h$ are differentiable functions on $(a,b)$ that are continuous on $[a,b]$. Define

:$D(x)\; =\; \backslash begin\{vmatrix\}$

f(x) & g(x) & h(x)\\

f(a) & g(a) & h(a)\\

f(b) & g(b) & h(b)

\end{vmatrix}

There exists $c\backslash in(a,b)$ such that $D\text{'}(c)=0$.

Notice that

:$D\text{'}(x)\; =\; \backslash begin\{vmatrix\}$

f'(x) & g'(x)& h'(x)\\

f(a) & g(a) & h(a)\\

f(b) & g(b)& h(b)

\end{vmatrix}

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 $c\backslash in\; (a,b)$ such that $D\text{'}(c)=0$.

Let X and Y be non-negative random variables such that E[X] < E[Y] < ∞ and $X\; \backslash leq\_\{st\}\; Y$ (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

:$f\_Z(x)=\{\backslash Pr(Y>x)-\backslash Pr(X>x)\backslash over\; \{\backslash rm\; E\}[Y]-\{\backslash rm\; E\}[X]\}\backslash ,,\; \backslash qquad\; x\backslash geqslant\; 0.$

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 y ≥ x ≥ 0. Then, E[g′(Z)] is finite and{{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 |s2cid=250351233 }}

:$\{\backslash rm\; E\}[g(Y)]-\{\backslash rm\; E\}[g(X)]=\{\backslash rm\; E\}[g\text{'}(Z)]\backslash ,[\{\backslash rm\; E\}(Y)-\{\backslash rm\; E\}(X)].$

{{See also|Voorhoeve index|Mean value problem}}

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 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

:$\backslash operatorname\{Re\}(f\text{'}(u))\; =\; \backslash operatorname\{Re\}\backslash left\; (\; \backslash frac\{f(b)-f(a)\}\{b-a\}\; \backslash right),$

:$\backslash operatorname\{Im\}(f\text{'}(v))\; =\; \backslash operatorname\{Im\}\backslash left\; (\; \backslash frac\{f(b)-f(a)\}\{b-a\}\; \backslash right).$

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

• Newmark-beta method
• Mean value theorem (divided differences)
• Racetrack principle
• Stolarsky mean
• Mean value problem

{{Reflist|50em}}

• {{cite book | last=Rudin | first=Walter | title=Principles of Mathematical Analysis | publisher=McGraw-Hill Publishing Company | publication-place=Auckland | date=1976 | isbn=978-0-07-085613-4}}
• {{citation|title=The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis|series=Classics in Mathematics|first=Lars|last= Hörmander|authorlink=Lars Hörmander|publisher=Springer|year= 2015|edition=2nd|

isbn= 9783642614972}}

• {{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

{{Calculus topics}}

{{Joseph-Louis Lagrange}}

{{Authority control}}

Category:Augustin-Louis Cauchy

Category:Articles containing proofs

Category:Theorems in calculus

Category:Theorems in real analysis