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draw points of convex function on [0,1] (VII Soros Olympiad 2000-01 R1 10.3)

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July 28, 2021
functionalgebra

Problem Statement

Let y=f(x)y = f (x) be a convex function defined on [0,1][0,1], f(0)=0,f (0) = 0, f(1)=0f (1) = 0. It is also known that the area of ​​the segment bounded by this function and the segment [0,1][0, 1] is equal to 11. Find and draw the set of points of the coordinate plane through which the graph of such a function can pass. (A function is called convex if all points of the line segment connecting any two points on its graph are located no higher than the graph of this function.)
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