Cauchy's Mean-Value Theorem
Cauchy's mean-value theorem is a generalization of the usual mean-value theorem. It states that if 
and 
are continuous
on the closed interval ![[a,b]](/National_Library/20160521004321im_/https://mathworld.wolfram.com/images/equations/CauchysMean-ValueTheorem/Inline3.gif)
, if 
, and
if both functions are differentiable on the open interval 
, then there
exists at least one 
with 
such
that
 |
(Hille 1964, p. 340).
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Bolzano's theorem
