Second Derivative Test for Constrained Extrema

This handout presents the second derivative test for a local extrema of a Lagrange multiplier problem. The Section 1 presents a geometric motivation for the criterion involving the second derivatives of both the function f and the constraint function g. The main result is given in section 3, with the special cases of one constraint given in Sections 4 and 5 for two and three dimensions respectively. The result is given in terms of the determinant of what is called the bordered Hessian matrix, which is defined in Section 2 using the Lagrangian function.