In real analysis, the mean value theorem for differentiation states the following:
The formula ( f(b) - f(a) ) / (b - a) gives the slope of the line joining the points (a , f(a)) and (b , f(b)), which we call a chord of the curve, while f ' (x) gives the slope of the tangent to the curve at the point (x , f(x) ). Thus the Mean value theorem says that given any chord of a smooth curve, we can find a tangent parallel to that chord and moreover we can take the tangent to some point lying between the end-points of the chord.
The mean value theorem can be used to prove Taylor's theorem, of which it is a special case.
Proof of the theorem: Define g(x) = f(x) + rx , where r is a constant. Since f is continuous on [a , b] and differentiable on (a , b), the same is true of g. We choose r so that g satisfies the conditions of Rolle's theorem, which means
Generalization: The theorem is usually stated in the form above, but it is actually valid in a slightly more general setting: We only need to assume that f : [a , b] -> R is continuous on [a , b], and that for every x in (a , b) the limit limh->0 (f(x+h)-f(x))/h exists or is equal to +/- infinity.
The first mean value theorem for integration states:
The second mean value theorem for integration states: