A Unique Area Property of the Quadratic Function

摘要

Suppose that a function f defined on the real line is convex or concave with f"(x) continuous and nonzero for all x. Let (x_1, f(x_1)) and (x_2, f(x_2)) be two arbitrary points on the graph of f with x_1 < x_2. For i = 1, 2, let L_i denote the tangent line to f at the point (x_i, f(x_i)) and let A_i be the area of the region R_i bounded by the graph of f, the tangent line L_i, and the line x = x, the x-coordinate of the intersection of L_1 and L_2. It is proved that f is a quadratic function if and only if A_1 = A_2 for every choice of x_1 and x_2.