 Solution

At a regular constrained extremum of $f$ on $g=0$, the tangent derivatives of $f$ vanish. Since $\nabla g$ is normal to the constraint surface, the Lagrange multiplier condition isOne solves these equations and then compares the resulting candidates, including any boundary or singular cases.

Let the base have dimensions $x$ and $y$, and let the height be $z$. Measure cardboard relative to the thickness of the front and back. The weighted amount used isbecause the bottom has triple thickness, the two front and back faces have ordinary thickness, and the two side faces have double thickness. The constraint is $xyz=3$.

The multiplier equations areThey implyThus $x=2y$ and $z=3y/2$; imposing $xyz=3$ gives $y=1$. ThereforeThis is the global minimum: by the arithmetic-geometric mean inequality,and equality holds at these dimensions. This is an instance of weighted open-box minimization.