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Cauchy Mean-Value Theorem

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The Cauchy mean-value theorem states that if
f
and
g
are two functions continuous on
[a,b]
and differentiable on
(a,b)
, then there exists a point
c
in
(a,b)
such that
f'(c)(g(b)-g(a))=g'(c)(f(b)-f(a))
.
Geometric interpretation: Consider the parametric curve
X(t)=(f(t),g(t))
,
t∈[a,b]
,
X(a)≠X(b)
; then the line passing through
X(a)
,
X(b)
is parallel to the tangent line passing through
X(c)
.
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