$X>0$, $Y>0$ and $X^2+Y ^3\geq X^3+Y^4$. Prove that $X^3+Y ^3\leq 2$

First I tried:

$0<X\leq1$ and $0<Y\leq1$, but this in not the only case for $X^2+Y ^3\geq X^3+Y^4$ and get nowhere. 

Can somobody give me an idea?