Lagrange Multipliers: Two Constraints

With one constraint, the gradient of the objective function is parallel to a single constraint gradient. With two constraints, the gradient of the objective function becomes a linear combination of two constraint gradients, forming a plane of normal directions. This allows optimization along the curve where both constraint surfaces intersect, rather than just on a single surface.

Each constraint forms a surface in three-dimensional space. Feasible points are those satisfying both constraints simultaneously, and they typically lie along the curve where the two constraint surfaces intersect. The optimization problem is restricted to finding the maximum or minimum only among points on this intersection curve, not across all space.

At an optimal point, the objective function cannot increase or decrease along the feasible curve, so its gradient must be perpendicular to the curve's tangent direction. The two constraint gradients are perpendicular to their respective surfaces and together define a plane. The objective gradient lies in this plane, making it expressible as a linear combination of the two constraint gradients.

Lambda and mu are scalar multipliers that express the objective function's gradient as a linear combination of the two constraint gradients. They appear in the partial derivative equations that form part of the complete system. Solving for these multipliers alongside the variables yields candidate points for the constrained optimum.

Partial derivatives are taken with respect to each variable, incorporating the two multipliers. These derivative conditions are combined with the two original constraint equations to form a complete system. Solving this system simultaneously gives the values of all variables and multipliers, identifying candidate points for the constrained maximum or minimum.

The surface area of the carton is the objective function being minimized to reduce material costs. The two constraints are the fixed volume requirement and the shipping size requirement. The optimal carton design must satisfy both constraints while achieving the minimum surface area.

After solving the system, candidate points are identified. These points must then be checked in the context of the problem to determine which represents the required maximum or minimum. This verification ensures the solution is physically meaningful and satisfies all problem requirements.