Lecture 5. Local Extremum of a Function of Two Variables

This lecture provides a foundational introduction to finding local extrema (maxima and minima) for functions of two variables. It begins by clearly defining what constitutes a local maximum and minimum. The material then guides students through the essential mathematical conditions for identifying these points.

The lecture covers the necessary first-order condition, which helps locate stationary points, and the more powerful sufficient second-order condition. This involves analyzing second derivatives and the Hessian determinant to determine whether a stationary point is a maximum, a minimum, or a saddle point. A clear, step-by-step algorithm is presented to streamline the investigation process. The theory is reinforced with several worked examples, including a special case where the standard test is inconclusive, demonstrating a complete approach to function analysis.