A Consumer seeks to maximize a utility function U(X,Y)subject to his income constraint given by:P₁X + P₂Y = M.

a)What is meant by a duality problem in constrained optimization? Please provide examples.

b)Set up a Lagrange function for this optimization.

c)State the First order conditions and explain how you would solve for the critical values.

d)Explain the meaning of the Lagrange multiplier as it related to this optimization problem.

e.)Given a utility function: U(X,Y) = X^1/2Y^1/2 and an income constraint: 50 = 3X + 2Y. maximize the utility function, subject to the constraint