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

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).
Old Node ID: 
2644
MSC Codes: 
Author(s): 
J. Aczel
Publication Date: 
Monday, July 14, 2008
Original Publication Source: 
Mathematics Magazine
Original Publication Date: 
January, 1985
Subject(s): 
Calculus
Single Variable Calculus
Differentiation
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Author (old format): 
J. Aczel
Applicable Course(s): 
4.11 Advanced Calc I, II, & Real Analysis
Modify Date: 
Monday, July 14, 2008
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