A family of functions: x² + m/x

Case $m\le0$. $2x^3-m>0$ for $x>0$: $f_m$ is strictly increasing on $(0,\infty)$. It has no minimum: for $m<0$, $f_m(x)\to-\infty$ as $x\to0^+$; for $m=0$, $f_0(x)=x^2$ has infimum $0$ which is not attained on $(0,\infty)$.

Case $m>0$. $f_m'$ vanishes at $x_0=\left(\dfrac m2\right)^{1/3}$, is negative before and positive after: $f_m$ decreases on $(0,x_0]$ and increases on $[x_0,\infty)$. Since $m=2x_0^3$,
$$f_m(x_0)=x_0^2+\frac{2x_0^3}{x_0}=3x_0^2=3\left(\frac m2\right)^{2/3}.$$
This is the global minimum. Also $f_m\to+\infty$ at both ends.

Application. For $m=16$: $x_0=2$ and the minimum is $f(2)=4+8=12=3\cdot8^{2/3}$.

Remark. The qualitative behaviour changes at $m=0$, a bifurcation value: for $m>0$ the term $\frac mx$ creates a minimum, for $m\le0$ it does not.