Множители Лагранжа: два ограничения

Consider a rectangular shipping carton with dimensions x, y, and z. The goal is to minimize the surface area to reduce material costs. At the same time, the carton must meet two conditions: a fixed volume and a shipping size requirement.

In this problem, surface area is the quantity being minimized, while volume and shipping size are the constraints.

Geometrically, each constraint forms a surface in three-dimensional space. Any carton design that meets both constraints must lie on the curve where these two surfaces intersect.

To find the optimal point on this curve, the method of Lagrange multipliers with two constraints is used. This introduces two multipliers, lambda and mu. At this point, the gradient of the surface area is a linear combination of the gradients of the two constraints. This means the gradient of the surface area lies in the plane spanned by the two constraint gradients.

Next, partial derivatives with respect to each variable are computed. These derivative equations, together with the two constraint equations, form a system. Solving this system gives the values of x, y, and z that minimize the surface area while meeting both requirements.