11. Consider a consumer who chooses a bundle of goods to maximize utility, subject to an affordability constraint: . U(x, y) = xy+2x+12y I = Pxx+ Pry . 1 = 100; Px = 2; Py = 4 a. Write down the Lagrangian function, L = L(x, y, 1), for this problem and differentiate it with respect to x, y, and A. b. Write down the first order necessary conditions for a constrained max or min of the utility function and solve them for the utility maximizing combination of quantities for good X and good Y: x* and y*. c. What is the Lagrange multiplier (2*) associated with the utility-maximizing bundle? d. Use the Hessian matrix of the Lagrangian function's second derivatives, H(L), to verify that you have, indeed, identified a constrained maximum point