Eco 429;

(6) Consider the function f(x, y. 2) = (x - 1)3 + (y - 2)' + (2 - 3)2 and the set S = {(x, y.=):cty+ = =1} (a) Write the Lagrangian function and the Lagrange equations. 5 points (b) Using the Lagrangian method, find the minima of f(x, y, 2) on the set S. 5 points(5) Consider the function f(x, y) = xy- a - yz (a) Determine the extreme points of f in the set R?. 5 points (b) Classify the extreme points of f in the set R and justify if they are local or global extreme points. 5 points(4) Consider the set of equations ry + zy = 1 cty-2 0 (a) Prove that the above system of equations determines implicitly two differentiable functions y(x) and z(x) in a neighborhood of the point (r, y, z) = (0, 1, 1). 4 points(b) Compute y' (0), ='(0) and the first order Taylor polynomial of y(x) and s(x) at the point co = 0. 6 points(3) Consider the function f(x, y) = ar' + ay? - 2ry where a ( R is a parameter and a > (). (a) Compute the gradient and the Hessian matrix of the function f. Compute the Taylor polynomial of degree 2 of f, centered at the point r = 0, y = 1. Compute the critical points of f. 5 points