Mean value theorem

The mean value theorem of real analysis states the following:

Let f : [a , b] -> R be continuous on the interval [a , b] and differentiable on (a , b). Then there exists some x in (a , b) with

f '(x) = ( f(b) - f(a) ) / (b - a)

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.

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

f(a) + ra = f(b) + rb

=> r = -( f(b) - f(a) ) / (b - a)

By Rolle's Theorem, there is some x in (a , b) for which g '(x) = 0, and it follows

f '(x) = g '(x) - r = 0 - r = ( f(b) - f(a) ) / (b - a)

as required.