Vrftjkry

up vote 2 down vote favorite The questions has two parts: Prove (i) $ xy^{3} leq frac{1}{4}x^{4} + frac{3}{4}y^{4} $ and (ii) $ xy^{3} + x^{3}y leq x^{4} + y^{4}$ . Now then, I went about putting both sides of $sqrt{xy} leq frac{1}{2}(x+y)$ to the power of 4 and it left me with $$-x^{3}y leq frac{1}{4}x^{4} + frac{1}{4}y^{4} + xy^{3} + frac{5}{2}x^{2}y^{2}. $$ Curiously squaring and multiplying $sqrt{xy} leq frac{1}{2}(x+y)$ I've tried merging the results with my other inequality a few times to no avail - I just can't seem to get the signs right and nor can I seem to make the coefficients of the $x^{4}$ and $y^{4}$ different, as they are in (i). I feel like there's something I'm missing. Does anyone see a nice way about this problem? inequality a.m.