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A Unique Area Property of the Quadratic Function

Connie Xu
Mathematics Magazine
Vol. 87, No. 1 (February 2014), pp. 52-56
DOI: 10.4169/math.mag.87.1.52
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Page Count: 5
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A Unique Area Property of the Quadratic Function
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Summary Suppose that a function f defined on the real line is convex or concave with f″(x) continuous and nonzero for all x. Let (x1, f(x1)) and (x2, f(x2)) be two arbitrary points on the graph of f with x1 < x2. For i = 1, 2, let Li denote the tangent line to f at the point (xi, f(xi)) and let Ai be the area of the region ℜi bounded by the graph of f, the tangent line Li, and the line x = x, the x-coordinate of the intersection of L1 and L2. It is proved that f is a quadratic function if and only if A1 = A2 for every choice of x1 and x2.

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