# Cauchy Mean-Value Theorem

Cauchy Mean-Value Theorem

The Cauchy mean-value theorem states that if and are two functions continuous on and differentiable on , then there exists a point in such that .

f

g

[a,b]

(a,b)

c

(a,b)

f'(c)(g(b)-g(a))=g'(c)(f(b)-f(a))

Geometric interpretation: Consider the parametric curve , , ; then the line passing through , is parallel to the tangent line passing through .

X(t)=(f(t),g(t))

t∈[a,b]

X(a)≠X(b)

X(a)

X(b)

X(c)