(1) Consider the following system of linear equations 3x + 2y + 203 = -13 -5 5x + 2ay - 3: = -18 where a E R is a constant. (a) Classify the system according to the values of a b) Solve the above linear system for the values of a for which the system has infinitely many solutions. How many parameters are needed to describe the solution ? (2) Consider the function f : R' - R. (a) Write the definition that f is continuous at the point p = (ro, Do). b) Consider the function f(1. y) = 3+7 if (x, y) # (0.0), if (I, y) = (0, 0). Determine if the function f is continuous at the point (0,0). (3) Consider the function f(x, y) = 2reW/x the point p = (2,0) and the vector v = (3,2). (a] Compute the gradient of the function f at the point p. Compute the tangent plane to the graph of f at the point (2, 0.4). b) Compute the directional derivative of the function / at the point p in the direction of the vector v. Which is the direction of maximal growth of f at the point p? Which is the maximum value of the directional derivative? (4) Consider the function f(x,y.=)= x - y' +2ry - - +6 and the set A = [(I, M. =) ( R] : = = 2r +2y-1} a) Compute the Lagrange equations that determine the extreme values of f on A. b) Determine the points that satisfy the Lagrange equations. (5) Consider the following maximization problem max y(x - 1) s.t. (1 - 1)' +y's2 (a) Compute the Kuhn-Tucker equations that determine the extreme values of f on A. "b) Determine the points that satisfy the Kuhn-Tucker equations.(1) Consider the following system of linear equations :+ (k+1)y +2: -1 katy+= k (k - 1) x - 2y-2 = 4+1 where k E R is a constant. [a) Classify the system according to the values of k. b) Solve the above system for the values of & for which the the system has infinitely many solutions. How many parameters are needed to describe the solution? (2) Consider the function /(z,y) = 3xin(2' - y), the point p = (2,3) and the vector v = (1,2). (a) Find the directional derivative of the function f at the point p in the direction of the vector v. b) What is the direction of maximal growth of f at the point p? What is the maximal value of the directional derivative of f at the point p? (3) Consider the function / : R' - R f(z,y) = to if (z, y) # (0,0), if (z, y) = (0,0). (a) Study if the function f is continuous at the point (0,0). Study at which points of R? the function f is continuous. (b) Argue in which of the following sets we may apply the Theorem of Welerstrass to show that the above function f attains a global extremum on the set A= ((ry) ER' : (x- 1)' +y's4) B = ((ry)ER :+ 21) C= ((ry) ER' : (x-3) +y's1) (4) A company sells two goods A and B. The total profits obtained from the sale of art units of A and ra units of B is the following: I(m, m) = -x - 3.r; - 2rian + 4x1 + 8x2 (m and ag in thousands of units) (a) Find the amounts in and ra which maximize the profits. b) Argue why can we be sure that the amount obtained in part (a) is a global maximum of the function (5) Consider the following maximization problem (x - 1) - V A.M S.L. -2x + 1/ $ 2 S+155 [a) Find the Kuhn-Tucker equations that determine the extreme points of f and draw the feasible set, (b) Which restrictions are binding at the point A = (1.0)7 Show that the point A = (1, 0) verifies the Khun-Tucker equations for the problem above