Consider an open box with no top, as shown. The box has volume $32$ and dimensions $x,y,z$. Using Lagrange multipliers, find the dimensions of the box with minimal surface area.
Solution
We observe this is a constrained optimization problem: we are to minimize surface area under the constraint that the volume is 32.
We now express the problem in terms of the variables $x,y,z$.
Because the box is open, it has surface area $2xy + 2xz + yz$.
Because these equations are nonlinear, we can not use linear algebra techniques to solve them. Instead, we try to make progress by writing equations for $\lambda$: \begin{align} \lambda &= \frac{2y+2z}{yz} \\ \lambda &= \frac{2 x+z}{xz} \\ \lambda &= \frac{2 x+y}{xy} \\ \end{align}
These equations can be simplified so that each variable only appears once: