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6. a) O Construct the Lagrangian function and state the necessary optimality conditions for the problem: optimize f(x,y) = x + 3xy + y - x + 3y Subject to x-y=-1 [4 marks) (ii) obtain the optimal points to the problem in (i) using the Lagrange method and 16 investigate the nature of the optimal points using sufficient conditions marks) b) () State the Karush-Kuhn-Tucker (KKT) necessary conditions for the problem: min(x-3)(x-2) Subject to so 30 [4 marks] (1) Hence obtain the optimal solution to the problem. 16 arks] 6. a) O Construct the Lagrangian function and state the necessary optimality conditions for the problem: optimize f(x,y) = x + 3xy + y - x + 3y Subject to x-y=-1 [4 marks) (ii) obtain the optimal points to the problem in (i) using the Lagrange method and 16 investigate the nature of the optimal points using sufficient conditions marks) b) () State the Karush-Kuhn-Tucker (KKT) necessary conditions for the problem: min(x-3)(x-2) Subject to so 30 [4 marks] (1) Hence obtain the optimal solution to the problem. 16 arks]