Suppose $x,y$ are positive real numbers that satisfy $$xy(x+2y)=2$$ What is the minimum value of $x+y$?
My Thoughts
I’ve attempted using arithmetic-geometric mean inequality and got:
$\frac{x+y+x+2y}{3} \geq \sqrt[3]{2}$
Therefore $2(x+y)+y \geq 3\sqrt[3]{2}$, then I got trapped.
Feels like I’m in the wrong way, I need a hint.
Best Answer
Let $t=x+y$ and we need $t_{\min}$. Then we have $$(t+y)(t-y)y=2\implies t^2y-y^3=2$$ and thus $$t^2 = y^2+{2\over y}$$ so if we apply Am-Gm for three terms we get $$t^2=y^2+{1\over y}+{1\over y}\geq 3$$
and minimum value $t=\sqrt{3}$ is achieved iff $y^2 = {1\over y}$ i.e. $y=1$ and $x=\sqrt{3}-1$.