There are five questions;

(1) Given the following system of linear equations with a parameter a ( R of+y+2 = stay+z = cty+oz = (a) Classify the system according to the values of a. |1 point (b) Solve the above system for the value of the parameter a = 1. 1 point (2) (a) Show that the following system of equations 13+2-r'+2 = 0 yz + 12- xy-1 = 0 defines two functions y = y(x), = = =(r) in a neighborhood of the point (x, y, =) = (2, 1, 1). Com- pute y'(2), = (2). 1 point (b) Consider the functions F(r, y, =) = xz-y' and G(x) = F(r, y(r), =(x)). Compute G'(2). |1 point (3) Consider the function /(r, y) = = + ay' with a e R defined on the open set D = {(x, y) ( R' : y > 0}. (a) Study the convexity of the function f in the set A, depending on the values of the parameter a. 1 point (b) Compute the Taylor polynomial of degree 2 of the function f at the point p = (0, 1). 1 point(4) Consider the function /(r, y. =) = x-y+z and the set A = {(x. y, =) ( R3 : ' +y-+2' =9,r+= = 4). (a) Compute the Lagrange equations that determine the extreme points of the function f in the set A. Compute the points that satisfy the Lagrange equations and the values of the corresponding Lagrange multipliers at each of the points. | 1 point (b) Using the second order conditions, classify the solutions found in the previous part into maxima, minima and local points. Can you say if any of the local maxima and/or maxima is a global extreme point on the set A? Justify adequately your answers. 1 point (5) Consider the function /(r, y) = r* +y) - 2a'r' - 3y with a c R, a # 0. (a) Determine the critical points of the function f in the set IR?. 1 point (b) Classify the critical points of the previous part into (local and/ or global) maxima and saddle points. 1 point