Finding lowest possible value please help me with this problem:
Find the lowest possible value of 
$$
x+y^3
$$
where both x and y are positive and x*y=1.
I know how to solve this one using my method, but I was suggested to use geometric and arithmetic mean. I have no idea how to solve this using them.

Since $xy = 1$, observe that $x + y^3 = y^{-1} + y^3 = y^3 + \frac{y^{-1}}{3} + \frac{y^{-1}}{3} + \frac{y^{-1}}{3} \geq 4\sqrt[4]{\frac{1}{3^3}}$.
The equality holds if and only if $y^3 = \frac{y^{-1}}{3}$, that means for $y = \frac{1}{\sqrt[4]{3}} > 0$, therefore your lowest possible value of $x + y^3$ is $4\sqrt[4]{\frac{1}{3^3}}$.
