Classify Critical Points Using Hessian Eigenvalues
Calculus & Optimization DS practice problem on Onlearn.
Difficulty: hard.
Topics: Understanding the Second-Order Sufficient Condition for local extrema in multivariable functions., Second-Order Partial Derivatives, Symmetry of Mixed Partials (Clairaut's Theorem), Characteristic Polynomials, Definiteness of Quadratic Forms, Stationary Point Identification, Multivariable Calculus, Linear Algebra, Optimization Theory, Numerical Analysis, Matrix Theory, Partial Derivatives, Gradient Vector, Hessian Matrix, Eigenvalue Decomposition, Sylvester's Criterion.
Given a scalar valued function f(x, y) = x^3 + y^3 3xy, identify the critical points and classify them using the eigenvalues of the Hessian matrix. The function should return a dictionary mapping each critical point coordinate tuple to its classification string ('local minimum', 'local maximum', 'saddle point', or 'undefined').