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[1] Is there an easy way to formally prove that, $$ 2xy^{2} +2x^{2} y-2x^{2} y^{2} -4xy+x+y\ge -x^{4} -y^{4} +2x^{3} +2y^{3} -2x^{2} -2y^{2} +x+y$$ $${0 without resorting to checking partial derivatives of the quotient formed by the two sides, and finding local maxima?

[2] Similarly, is there an easy way for finding $$\max_{0 where, $$f(x,y)=2x(1+x)+2y(1+y)-8xy-4(2xy^{2} +2x^{2} y-2x^{2} y^{2} -4xy+x+y)^{2}$$

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    Wolframalpha suggests that there the maximum is attained in two points: try [maximize 2x(1+x)+2y(1+y)-8xy-4(2xy^2 +2x^2 y-2x^2 y^2 -4xy+x+y)^2 for 0<=x<=1, 0<=y<=1](http://www.wolframalpha.com/input/?i=maximize+2x%281%2Bx%29%2B2y%281%2By%29-8xy-4%282xy%5E2+%2B2x%5E2+y-2x%5E2+y%5E2+-4xy%2Bx%2By%29%5E2+for+0%3C%3Dx%3C%3D1%2C+0%3C%3Dy%3C%3D1) and [maximize 2x(1+x)+2y(1+y)-8xy-4(2xy^2 +2x^2 y-2x^2 y^2 -4xy+x+y)^2](http://www.wolframalpha.com/input/?i=maximize+2x%281%2Bx%29%2B2y%281%2By%29-8xy-4%282xy%5E2+%2B2x%5E2+y-2x%5E2+y%5E2+-4xy%2Bx%2By%29%5E2)2011-12-13

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