Mean Value Theorem:
If f(x) is continuous on the closed interval a x b and differentiable on the open interval a < x < b, some c between a and b (a < c < b) satisfies
Geometricaly, this is the statement that the secant to the curve f between a and b is parallel to some tangent line between them.
We can rewrite the mean value theorem as follows.
Setting h = x and c = y, we get
f '(y)(x-a) = f(x) - f(a)
f(x) = f(a) + f '(y)(x - a), for some y between x and a.
The theorem has a number of useful consequences:
Theorem 1: If f ' > 0, then f is increasing (that is if x1 < x2 , then f(x1) < f(x2)). Proof of 1
Theorem 2: If f ' = 0, then f is constant. Proof of 2
Corollary: If f ' = g', then f = g + c where c is independent of x: c' = 0. Proof of corollary