Lagrange'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) = f(b)-f(a)/b-a.
Proof:-
Let us define a function
F(x) = f(x)(b-a)-[f(b)-f(a)]x........(i)
which 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)(b-a)-[f(b)-f(a)]a
= bf(a)-af(b)-af(b)+af(a)=bf(a) - af(b)

F(b) = f(b)(b-a)-[f(b)-f(a)]b
= bf(b)-af(b)-bf(b)+bf(a)
=bf(a)-af(b) = F(a)

The by Rolle's Theorem there exist at least one point c in (a,b) such that F'(c) = 0.
But F'(c) = f'(c)(b-a)-[f(b)-f(a)].
f'(c)(b-a)-[f(b)-f(a)] = 0
f'(c) = f(b)-f(a)/b-a