Try to break y^3+y into three terms in order to use A-G inequality for 6 terms
y^3+y >= (y^3+2y^2+y)/2
The original left >= 6*xy*(1/4)^{1/6}
6*(1/4)^{1/6}>9/2
y^3+y >= (y^3+2y^2+y)/2
The original left >= 6*xy*(1/4)^{1/6}
6*(1/4)^{1/6}>9/2
