[latex]sqrt{ x^{2} + y^{2} }>sqrt[3]{ x^{3} + y^{3} } /()^6[/latex] [latex](x^2 + y^2)^3 > (x^3 + y^3)^2[/latex] [latex]x^6 + 3x^4y^2 + 3x^2y^4 + y^6 > x^6+2x^3y^3+y^6[/latex] [latex]x^6 + 3x^4y^2 + 3x^2y^4 + y^6- x^6-2x^3y^3-y^6>0[/latex] [latex]3x^4y^2 - 2x^3y^3 + 3x^2y^4>0[/latex] [latex]x^2y^2(3x^2 - 2xy + 3y^2)>0[/latex] ------------- [latex]3x^2 - 2xy + 3y^2=3x^2-2xy+ frac{1}{3}y^2+ frac{8}{3}y^2=[/latex] [latex]3left(x^2- frac{2}{3} xy+ frac{1}{9}y^2 ight) + frac{8}{3}y^2=[/latex] [latex]3left(x- frac{1}{3}y ight)^2 + frac{8}{3}y^2[/latex] ------------- [latex]underbrace{x^2y^2}_{>0}left[ 3underbrace{left(x- frac{1}{3}y ight)^2 }_{>0}+underbrace{frac{8}{3}y^2}_{>0} ight]>0[/latex]
