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)

.