应用优化问题

Translation before differentiation

A worked packing problem

A worked Lagrange problem

Reporting with justification

A closed box has volume $32$ with side lengths $x, y, z$. Which translation into objective and constraint is correct for minimizing surface area?

Minimizing the surface of the closed box with volume $32$ produces which optimal shape?

In the production problem, the multiplier at the optimum is $\lambda = 12.5$. What does it report?

In the production problem with $P = xy$ and budget $4x + y = 100$, the optimum spends $d_1 = 4x$ on the first input and $d_2 = y$ on the second. Give the value of $d_1$ at the optimum, as a plain number.

At the optimal cube candidate for the box problem, the Hessian check yields the discriminant $D = 12$. Give the sign of $f_{xx}$ there, entering + or - as the word plus or minus.