Proof
The expression (f(b) − f(a)) / (b − a) gives the slope of the line joining the points (a, f(a)) and (b, f(b)), which is a chord of the graph of f, 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 point lying between the end-points of the curve such that the tangent at that point is parallel to the chord. The following proof illustrates this idea.
Define g(x) = f(x) − rx, where r is a constant. Since f is continuous on and differentiable on (a, b), the same is true for g. We now want to choose r so that g satisfies the conditions of Rolle's theorem. Namely
By Rolle's theorem, since g is continuous and g(a) = g(b), there is some c in (a, b) for which, and it follows from the equality g(x) = f(x) − rx that,
as required.
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