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:
“If any doubt has arisen as to me, my country [Virginia] will have my political creed in the form of a “Declaration &c.” which I was lately directed to draw. This will give decisive proof that my own sentiment concurred with the vote they instructed us to give.”
—Thomas Jefferson (1743–1826)
“The proof of a poet is that his country absorbs him as affectionately as he has absorbed it.”
—Walt Whitman (1819–1892)
“There are some persons in this world, who, unable to give better proof of being wise, take a strange delight in showing what they think they have sagaciously read in mankind by uncharitable suspicions of them.”
—Herman Melville (1819–1891)