For each of the following functions, write the Lagrangian function and the Kuhn-Tucker conditions (including writing out the derivatives). You do not need to solve for the critical values. For all parts of this problem, x and y cannot be negative.

1.Maximize z = 3xy, subject to 2x + 2y 20

2.Maximize z = x² + 2xy + y², subject to 2x + y 24 and x + 3y 8

3.Minimize z = x² + 2y², subject to 4x 3y 20 and 3x² 2y 15