1. Consider the problem of maximizing xy 2 on the constraint set x 2+y 2 = 4, x 0, y 0.

(a) State the non-degenerate constraint qualification (NDCQ). Does it always hold in the constraint set?

(b) Form the Lagrangian. You can use either the standard Lagrangian, which has multipliers for the nonnegativity constraints, or the Kuhn-Tucker form, which omits the multipliers for the non-negativity constraints.

(c) Write the first-order conditions.

(d) Use the first order conditions to characterize the solution(s), including the multiplier to the equality constraint. Be explicit about each step.

(e) Suppose the equality constraint changes to x 2+y 2 = 6. Without solving the problem under the new constraint, use the meaning of the multiplier to approximate the change in the objective function.

Please show all work, thanks! I will give thumbs up if all correct :)

1. Consider the problem of maximizing 1-y on the constraint set x2 + y2 = 4, 1 > 0, y > 0. (a) State the non-degenerate constraint qualification (NDCQ). Does it always hold in the constraint set? (b) Form the Lagrangian. You can use either the standard Lagrangian, which has multipliers for the nonnegativity constraints, or the Kuhn-Tucker form, which omits the multipliers for the non-negativity constraints. (c) Write the first-order conditions. (d) Use the first order conditions to characterize the solution(s), including the multiplier to the equality constraint. Be explicit about each step. (e) Suppose the equality constraint changes to r?+y? = 6. Without solving the problem under the new constraint, use the meaning of the multiplier to approximate the change in the objective function