You construct a rectangular Box with volume K cm^3 Prove that a cube uses the least amount of material to construct the box

Let $x$, $y$ and $z$ the lenghts of the sides of the box. If we suppose this box is closed, we must to minimize the function $f(x,y,z)=2xy+2xz+2yz$ restricted to $xyz=K$, we can write the volume of the box, $V$, as follows
$$V=2\left(xy+\frac{K}{y}+\frac{K}{x}\right)$$
If we fix $x$ then we have $V_y=2\left(x-\frac{K}{y^2}\right)$ and $V_{yy}=\frac{4K}{y^3}>0$ then $V$ reach its minimun value for this $x$ when $y=\sqrt{\frac{K}{x}}$. Now we are looking where the function $x\mapsto2\left(2\sqrt{Kx}+\frac{K}{x}\right)$ reach its minimum value.
