Practical Class 5. Higher-Order Derivatives. Local Extremum

This practical class focuses on computing second-order partial derivatives and investigating functions of two variables for local extrema. The material is intended for students studying mathematical analysis who are familiar with differential calculus basics.

The class begins with a brief review of theoretical concepts: second-order partial derivatives, Schwarz's theorem, the Hessian matrix, and necessary and sufficient conditions for extrema. The main part consists of problems with detailed solutions, ranging from basic to advanced difficulty. Examples include computing second derivatives, finding stationary points, applying the sufficient condition, and handling the special case where the Hessian determinant equals zero.

The class concludes with independent work and a homework assignment covering all topics. Recommendations for solving problems and an analysis of typical mistakes are provided.