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integral inequality

Source: IMC 1998 day 1 problem 6

November 1, 2005
calculusintegrationinequalitiesfunctiontrigonometryreal analysisreal analysis unsolved

Problem Statement

Let f:[0,1]→Rf: [0,1]\rightarrow\mathbb{R} be a continuous function satisfying xf(y)+yf(x)≤1xf(y)+yf(x)\le 1 for every x,y∈[0,1]x,y\in[0,1]. (a) Show that ∫01f(x)dx≤π4\int^1_0 f(x)dx \le \frac{\pi}4. (b) Find such a funtion for which equality occurs.
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