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?

We know that: $$(x^2+y^2)\ge 2xy$$
Multiplying both sides by $y^2$
$$(x^2+y^2)\cdot y^2\ge 2xy^3$$
And clearly $x^2+y^2 \ge y^2$. So
$$(x^2+y^2)\cdot(x^2+y^2)\ge(x^2+y^2)\cdot y^2\ge 2xy^3$$
implying that $$\boxed{(x^2+y^2)^2 \ge 2xy^3}$$
