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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