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

Proof

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)

or

f(x) = f(a) + f '(y)(x - a), for some y between x and a.

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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

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Application