Theorem: The Mean Value Theorem States that let a real valued function f be continous on a closed interval [a,b] and differentiable in the open interval (a,b) then there exists atleast a point c (a,b) such that f(b)-f(a)=(b-a)f'(c) 
PROOF: by Rolles Theorem let 


We see that L(a)=f(a) and  L(b)=f(b),
now let g(x)=f(x)-L(x). Then g Is continous on [a, b], and differentiable on (a, b) and satisfies the condition g(a)=g(b)=0, by Rolles Theorem there exists cε(a,b)  such that g'(c)=0, by differentiating we see that f'(c)=L'(c). But





So








QED