6.1.3 Exercises on optimization with an equality constraint: sufficient conditions for a local optimum for a function of two variables
- For the problem
maxx,y x2 + y2 subject to x2 + xy + y2 = 3find all the solutions of the first-order conditions and determine, if possible, whether each solution is a local maximizer or a local minimizer.The bordered Hessian is
0 2x + y x + 2y 2x+y 2 − 2λ −λ x + 2y −λ 2 − 2λ . - (1, 1, 2/3): determinant is −24 < 0, so (1, 1) is a local minimizer.
- (−1, −1, 2/3): determinant is −24 < 0, so (−1, −1) is a local minimizer.
- (√3, −√3, 2): determinant is 24 > 0, so (√3, −√3) is a local maximizer.
- (−√3, √3, 2): determinant is 24 > 0, so (−√3, √3) is a local maximizer.