Assess students' ability to apply Lagrange multipliers to find and classify constrained extrema of differentiable functions of several variables subject to one or more equality constraints. Tasks should include forming the Lagrangian, solving the system ∇f = Σλ_i∇g_i along with the constraints, identifying all critical points, and using the bordered Hessian or an appropriate second-derivative test to determine maxima, minima, or saddle behavior; include at least one problem with multiple solutions or a degenerate/edge case for analysis.