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Maximum of f_k(x,y) with x^2+y^2=1

Source: Austrian Federal Competition for Advanced Students 2011, #2

May 15, 2011
inequalitiesinequalities proposed

Problem Statement

For a positive integer kk and real numbers xx and yy, let f_k(x,y)=(x+y)-\left(x^{2k+1}+y^{2k+1}\right)\mbox{.} If x2+y2=1x^2+y^2=1, then determine the maximal possible value ckc_k of fk(x,y)f_k(x,y).
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