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.
Read more about this topic: Mean Value Theorem
Famous quotes containing the word proof:
“The chief contribution of Protestantism to human thought is its massive proof that God is a bore.”
—H.L. (Henry Lewis)
“He who has never failed somewhere, that man can not be great. Failure is the true test of greatness. And if it be said, that continual success is a proof that a man wisely knows his powers,—it is only to be added, that, in that case, he knows them to be small.”
—Herman Melville (1819–1891)
“If some books are deemed most baneful and their sale forbid, how, then, with deadlier facts, not dreams of doting men? Those whom books will hurt will not be proof against events. Events, not books, should be forbid.”
—Herman Melville (1819–1891)