(MC09) Lagrange Multipliers

We begin by revisiting the concept of gradients and tangent vectors, emphasizing their orthogonality to contours, which sets the stage for understanding Lagrange multipliers. This method is crucial for finding local maxima and minima of functions subject to constraints. After a conceptual introduction and derivation of Lagrange multipliers, we demonstrate their practical application through two detailed examples: optimizing on contours and within regions bounded by contours.
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Note that for the extrema at the origin, being a max or a min depends on direction, as one can dind its a saddle for the surface (unless you restrict the domain).