Mean value theorem

The Mean Value Theorem of real analysis states the following:
Let f:[a , b] -> R be continuous on [a , b] and differentiable on (a , b). Then, for some x in (a , b)
f '(x) = ( f(b) - f(a) ) / (b - a)

The formula ( f(b) - f(a) ) / (b - a) gives the gradient 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 gradient of the tangent to the curve at the point (x , f(x) ). Thus the MVT 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 MVT: Define g(x) = f(x) + hx , where h is a constant. Since f is continuous on [a , b] and differentiable on (a , b) the same is true of g. We choose h so that g satisfies the conditions of Rolle's Theorem. Then
f(a) + ha = f(b) + hb
=> h = -( f(b) - f(a) ) / (b - a)
By Rolle's Theorem, there is some x in (a , b) for which g ' (x) = 0.
i.e. f ' (x) = -h = ( f(b) - f(a) ) / (b - a) , as required