Editing Mean value theorem (section)

==Generalizations==
===Linear algebra===
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) = \begin{vmatrix}
f(x) & g(x) & h(x)\\
f(a) & g(a) & h(a)\\
f(b) & g(b) & h(b)
\end{vmatrix}</math>

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

Notice that
:<math> D'(x) = \begin{vmatrix}
f'(x) & g'(x)& h'(x)\\
f(a) & g(a) & h(a)\\
f(b) & g(b)& h(b)
\end{vmatrix}</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 [[determinant]]s 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>.

=== Probability theory===
Let ''X'' and ''Y'' be non-negative [[random variable]]s such that E[''X''] < E[''Y''] < ∞ and <math>X \leq_{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 |s2cid=250351233 }}</ref>

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

=== Complex analysis ===
{{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:<ref>1 J.-Cl. Evard, F. Jafari, A Complex Rolle’s Theorem, American Mathematical Monthly, Vol. 99, Issue 9, (Nov. 1992), pp. 858-861.</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 the interior of the line segment from ''a'' to ''b'' such that
:<math>\operatorname{Re}(f'(u)) = \operatorname{Re}\left ( \frac{f(b)-f(a)}{b-a} \right),</math>
:<math>\operatorname{Im}(f'(v)) = \operatorname{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.