Find minimum of $2x^3+x+\frac3{x^2}$ for $x>0$ Using AM-GM inequality we have:
$$2x^3+x+\frac3{x^2}=2x^3+x+\frac3{2x^2}+\frac3{2x^2}\geq 4\sqrt[4]{2x^3x\frac3{2x^2}\frac3{2x^2}}=4\sqrt[4]{\frac92}$$
As it turns out it is not the minimum, can you spot the mistake?

There is no flaw in your work - it is true that $$2x^3 + x + \frac{3}{x^2} \ge 4 \sqrt[4]{\frac{9}{2}}$$ for $x > 0$. However, AM-GM inequality won't be enough to find the minimum, as it only attains equality for $$2x^3 = x = \frac{3}{2x^2} = \frac{3}{2x^2}$$
which never happens.
