<s>[INST] How to prove $x^4+2x^2y^2+y^4\geq2xy^3$ Suppose that $x,y$ are real numbers. I want to prove $$x^4+2x^2y^2+y^4\geq2xy^3.$$ I noticed that this is the same as $$(x^2+y^2)^2\geq 2xy^3.$$ Can we proceed from here?
 [/INST] Consider :
$r>0$, $0\le \theta \lt 2π$.
Then obviously:
$r^4 \ge r\sin (2\theta) r^3 \sin^2 (\theta)$(Why?)
$r^4 \ge 2r \sin (\theta) \cos (\theta) r^3 \sin^2 (\theta)$
And with $x=r\cos (\theta)$, $y=r \sin (\theta)$:
$(x^2+y^2)^2\ge 2 xy^3.$
</s>