<s>[INST] If $x,y>0$ and $x^2+y^3\ge x^3+y^4$, prove that $x^3+y^3 \le 2$. As in the title. If $x,y>0$ and $x^2+y^3\ge x^3+y^4$, prove that $$x^3+y^3 \le 2.$$
This seems to be a very tricky one. I tried applying various inequalities like AM-GM, unfortunately, none of techniques I'm familiar with seem to work here.
I'd greatly appreciate any hints. 
 [/INST] By AM-GM $2x^3+1\geq3x^2$ and $3y^4+1\geq4y^3$.
Thus, $$2x^3+3y^4+2\geq3x^2+4y^3$$ or
$$x^3+y^3\leq2+3(x^3+y^4-x^2-y^3)\leq2$$
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