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

Start from a cube with sides $x$, $y$, and $z$ in length so $x=y=z$. The volume is $xyz$, and the surface area is $2(xy+xz+yz)$. To check if you can change this box so that it has the same volume but smaller area, scale one side by some number $c$ that is greater than 1. That means one of the sides must also scale, but by $1/c$. The surface area is now
$$2(x(cy)+x(\frac{1}{c})+yz)$$
And this is now larger than the previous area because
$$\begin{align*}
x(cy)+x(z/c)&=cxy+\frac{1}{c}xz\\
&=(x^2)(c+\frac{1}{c}),\,\,\,x^2=xy=xz=yz\\
&>x^2
\end{align*}
$$
And so you've now made the area $c+\frac{1}{c}$ times bigger. The same method can be shown for the scaling situation where both $x$ and $z$ scale downward in accordance with $y$ scaling upwards.
