Cauchy's Theorem

Statement:-
If f(x) is a continuous in a closed interval [a,b], differentiable in (a,b) then there exists at least one c in (a,b) such that f’(c)/g'(c) = f(b)-f(a)/g(b)-g(a).
Proof:-
Let us define a function
F(x) = f(x){g(b)-g(a)} - g(x){f(b)-f(a)}
This function is continious in [a,b] and differentiable in (a,b), since f(x) and x are continious in [a,b] and differentiable in (a,b).
Here F(a) = f(a){g(b)-g(a)} - g(a){f(b)-f(a)}
= f(a)g(b)-g(a)f(b)
F(b) = f(b){g(b)-g(a)} - g(b){f(b)-f(a)} = F(a)

The by Rolle's Theorem there exist at least one point c in (a,b) such that F'(c) = 0.
f'(c){g(b)-g(a)} = g'(c){f(b)-f(a)}
f’(c)/g'(c) = f(b)-f(a)/g(b)-g(a) g'(x) not equal to 0 in (a,b) and g(b) not equal to g(a)