Existence of a cone with minimal lateral area.

$A)$ Study the function $g(x)=\pi\cdot x\sqrt{x^2+\frac{2}{x^4}}$ and sketch its representative curve.

$B)$ We consider the set of right circular cones $E$ having the same volume $V=\frac{\pi \sqrt{2}}{3}$. Basing on the results of $A)$ prove that there exists, in $E$, at least a cone $(C_0)$ of minimal lateral area. Calculate in this case, the geometric characteristics of the cone $(C_0)$: the radius $R_0$ of the base, the height $H_0$ and the minimal lateral area $A_{1m}$.

I've already solved part $A)$ and I've skteched the curve, however I have no idea how to approach part $B)$ as I'm failing to see the relation between part $A)$ and $B)$.

Any help and hints are appreciated!

Let $x$ be the radius and $y$ be the height of some cone in $E$.
Its lateral surface area is $\pi x\sqrt{x^2+y^2}$.
Its volume is $\frac{1}{3}\pi x^2 y$, but they all have volume $\frac{\pi \sqrt{2}}{3}$, so they all have that certain $y$ such that $\frac{1}{3}\pi x^2 y=\frac{\pi \sqrt{2}}{3}$. Now plug that $y$ into the surface area to obtain $g$. Now you can connect the dots... (and use the derivative of $g$ to finish the question).