Suppose f(x) and g(x) are 2 functions satisfying three conditions:
1) f(x), g(x) are continuous in the closed interval a <= x <= b
2) f(x), g(x) are differentiable in the open interval a < x < b and
3) g'(x) != 0 for all x belongs to the open interval a < x < b
Then according to Cauchy’s Mean Value Theorem there exists a point c in the open interval a < c < b such that:
The conditions (1) and (2) are exactly same as the first two conditions of Lagranges Mean Value Theorem for the functions individually. Lagranges mean value theorem is defined for one function but this is defined for two functions.