Mean Value Theorem - Cauchy's Mean Value Theorem

Cauchy's mean value theorem, also known as the extended mean value theorem, is a generalization of the mean value theorem. It states: If functions f and g are both continuous on the closed interval, and differentiable on the open interval (a, b), then there exists some c ∈ (a,b), such that

Of course, if g(a) ≠ g(b) and if g′(c) ≠ 0, this is equivalent to:

Geometrically, this means that there is some tangent to the graph of the curve

which is parallel to the line defined by the points (f(a),g(a)) and (f(b),g(b)). However Cauchy's theorem does not claim the existence of such a tangent in all cases where (f(a),g(a)) and (f(b),g(b)) are distinct points, since it might be satisfied only for some value c with f′(c) = g′(c) = 0, in other words a value for which the mentioned curve is 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

which on the interval goes from the point (−1,0) to (1,0), yet never has a horizontal tangent; however it has a stationary point (in fact a cusp) at t = 0.

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 g(t) = t.