TITLE: Minimum of $x^2+\frac{a}{x}$ without Calculus
QUESTION [8 upvotes]: How can I find the minimum of $x^2+\frac{a}{x}$ on $\mathbb{R}_+$ without calculus?

REPLY [5 votes]: Revised (neater) answer: Fix $a>0$. For any $x>0$,
$$
\Big(x^2  + \frac{a}{x}\Big) - \Big(m^2  + \frac{a}{m}\Big) = \frac{{(x - m)(x^2 m + m^2 x - a)}}{{mx}} \geq 0,
$$
if $2m^3 = a$. We are done. (The above also shows that the minimum is strict.)