Editing Mean value theorem (section)

==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''', is a generalization of the 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>{{sfn|Rudin|1976|pp=107-108}} It states: if the functions <math>f</math> and <math>g</math> are both continuous on the closed interval <math>[a,b]</math> and differentiable on the open interval <math>(a,b)</math>, then there exists some <math>c \in (a,b)</math>, such that
[[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 <math>g(a) \neq g(b)</math> and <math>g'(c) \neq 0</math>, 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 \R^2\\t\mapsto (f(t),g(t))\end{cases}</math>

which is [[Parallel (geometry)|parallel]] to the line defined by the points <math>(f(a), g(a))</math> and <math>(f(b), g(b))</math>. However, Cauchy's theorem does not claim the existence of such a tangent in all cases where <math>(f(a), g(a))</math> and <math>(f(b), g(b))</math> are distinct points, since it might be satisfied only for some value <math>c</math> with <math>f'(c) = g'(c) = 0</math>, 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 \left(t^3,1-t^2\right),</math>

which on the interval <math>[-1,1]</math> goes from the point <math>(-1, 0)</math> to <math>(1, 0)</math>, yet never has a horizontal tangent; however it has a stationary point (in fact a [[cusp (singularity)|cusp]]) at <math>t = 0</math>.

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 <math>g(t) = t</math>.

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

Define <math>h(x) = (g(b)-g(a))f(x) - (f(b)-f(a))g(x)</math>, then we easily see <math>h(a)=h(b)=f(a)g(b)-f(b)g(a)</math>.

Since <math>f</math> and <math>g</math> are continuous on <math>[a,b]</math> and differentiable on <math>(a,b)</math>, the same is true for <math>h</math>. All in all, <math>h</math> satisfies the conditions of [[Rolle's theorem]]. Consequently, there is some <math>c</math> in <math>(a,b)</math> for which <math>h'(c) = 0</math>. Now using the definition of <math>h</math> we have:
:<math>0=h'(c)=(g(b)-g(a))f'(c)-(f(b)-f(a))g'(c)</math>
The result easily follows.