My favorite theorem is La grange's Mean Value Theorem . I like it because its quite simple but has many far fetched applications and is one of the most fundamental theorems of calculus. The theorem states that:
Let $ f(x) $ be a function of x subject to:
a. $ f(x) $ is a continuos function of x in the closed interval $ a<=z<=b $. b. $ f'(x) $ exists for every point in the open interval a<x<b, then there exists at least one value of x, say c such that $ a<c<b $ where $ f'(c)=\frac{f(b)-f(a)}{b-a} $.
Rolle's theorem is a special case of this theorem.
More information about when it was first described and its applications can be found here and also here .
