Assess students' ability to apply Lagrange multipliers to locate and classify extrema of multivariable functions subject to one or more smooth equality constraints. Tasks include setting up the system ∇f = Σλ_i∇g_i, solving algebraic systems (analytically or with guided numerical steps), handling multiple constraints, identifying and evaluating all candidate points (including relevant boundary cases), and using second-derivative tests or direct comparison to determine maxima, minima, or saddle behavior. Include interpretation of constraint geometry and recognition of degenerate cases.