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A Mean Value Property of the Derivative of Quadratic Polynomials-without Mean Values and Derivatives

by J. Aczel

This article originally appeared in:
Mathematics Magazine
January, 1985

Subject classification(s): Calculus | Single Variable Calculus | Differentiation
Applicable Course(s): 4.11 Advanced Calc I, II, & Real Analysis

Draw the secant line between any two points on the graph of a quadratic polynomial. The Mean Value Theorem tells us there is a point in that interval at which the derivative equals the slope of that line. That point, it turns out, is the midpoint of the line segment. This article proves, essentially, the converse -- that this midpoint property characterizes quadratic polynomials, and generalizes to the pure functional equation: [f(x)-f(y)] / (x-y)=h(x+y).

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