(1) Given the following system of linear equations with a parameter a c R stay += = a (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) (n) Show that the following system of equations 1+-x'+2 = 0 1 + 3: - xy- 1 defines two functions " = y(x), = = =(x) in a neighborhood of the point (r.y. =) = (2, 1, 1). Com- pute y'(2), ='(2). 1 point (b) Consider the functions F(x, y,=) = xz-y' and G(x) = F(x. y(2), =(x)). Compute G'(2). |1 point (3) Consider the function /(z, y) = = + ay' with a E R defined on the open set D= {(r, y) ER' : 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 /(z, y, =) = x -y+= and the set A = ((ry. =) ( R): s'+y'+2) =9,4+: =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 /(x.y) = at +y' - 2a 23 - 3y with a E R. a # 0. (a) Determine the critical points of the function f in the set R. 1 point (b) Classify the critical points of the previous part into (local and/or global) maxima and saddle points. 1 point(1) Consider the following system of linear equations with parameters a, be R ar+ += + (2a - 1 )y + = 2 - (a) Classify the system according to the values of a and b. |1 point (b) Solve the above system of linear equations for the values of the parameters a = b = 1. |1 point (2) Consider the function /(r.y) = x - y and the set A = {(r, y) ER' : -1 S y S (7,0 5 154). (a) Represent graphically the set A, its boundary, closure and interior. Justify if the function f attains a global maximum and/or a minimum in the set A, without computing them.|1 point (b) Using the level curves of the function f, compute the global extreme points of the previous part. 1 point (3) Consider the function f(x, y) = 347 if (I, y) # (0,0) if (z. y) = (0,0) (a) Compute the partial derivatives of f at the point (0, 0). |1 point (b) Determine if f is continuous and/or differentiable at the point (0, 0). 1 point (4) An agent has a utility function f(r, y,=) = Ina+2Ing +3Ins. The agent chooses the bundle (r. y.=) e R which maximizes his utility function f(z, y, =) on his budget line A = {(x, y. =) ( R' : 1 + 2y +3: = 90, r > 0,y > 0,=>0} (a) Write the Lagrange equations satisfied by the extreme points of f on the set A. Compute the points which satisfy the Lagrange equations and the value of the Lagrange multiplier, corresponding to each of those points. 1 point (b) Classify the solutions of the previous part into local maxima and minima, using the second order conditions. Can we say if any of the previous local maxima and minima is a global maximum or minimum in the set A? Justify the answers. |1 point (5) Consider the function /(r, y) = 10 - 8x + 2x' + 8y - Ary + bray + 2y" con b > 2. (a) Determine the critical points of the function f on the set R-.|1 point (b) Classify the critical points of f into (local or global) maxima, minima and saddle points, depending on the values of the parameter b. |1 point