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(1) Consider the following system of linear equations, ax - 2y +2 = I + ay + 2z = a where a E R is a parameter. (a) Classify the above system according to the values of the parameter a. (b) Solve the above system for the value of a for which the system is underdetermined. How many parameters are needed to describe the solution? (2) Consider the matrix (a) Find the characteristic polynomial and the eigenvalues of the matrix A. (b) For what values of the parameter a, is the matrix diagonalizable? (c) For the value a = 0, find the corresponding diagonal matrix and the matrix change of basis. (3) Given the linear map f : R3 - R4, f(x,y. =) = (x+y - z,2x + y + z,3x+ 2y, -x - 2=) (a) Write down the matrix of f (with respect to the canonical basis of R3 and R* ). Compute the dimensions of the kernel and the image of f. (b) Write a homogeneous system of equations that determines the kernel of f and a homogeneous system of equations that determines the image of f. What is the minimum number of equations necessary to describe each of these systems? (c) Find a basis of the image of f and a basis of the kernel of f. (4) Consider the set A = {(r, y) ( R' : r, y > 0; In(zy) 2 0). (a) Draw the set A, its boundary and its interior. Discuss whether the set A is open, closed, bounded, compact and/or convex. You must explain your answer. (b) Consider the function f(r, y) = x + 2y. Is it possible to use Weierstrass' Theorem to determine whether the function attains a maximum and a minimum on A? Draw the level curves of f, indicating the direction in which the function grows. (c) Using the level curves of f, find graphically (i.e. without using the first order conditions) if f attains a maximum and/or a minimum on A. (5) Consider the function f : R' - R f(z, V ) = NTV if (x, y) # (0,0) if (x, 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) Compute the partial derivatives of f at the point (0, 0), if they exist. Is the function f differentiable at the point (0, 0)? (6) Classify the quadratic form Q(x, y. =) = 3x2 + 3y + 5723 -Gary depending on the values of the parameter a. (7) Consider the function f(x, y) = x+y+ry+4 (a) Compute the critical points of f and classify them. (b) Determine the convex and open sets in R" where the function f is concave, and the convex and open sets in R where the function f is convex. (c) Determine if f attains any global extreme points on the set R2. (8) Consider the function f(x,y, =) = 4x+ 2y + = and the set A= ((x,y) ( R' : x +y' + :' = 21) (a) Find the Lagrange equations that determine the extreme points of f on the set A. (b) Determine the points that satisfy the Lagrange equations. (c) Using the second order conditions, classify the critical the points that satisfy the Lagrange equations.(5) Given the set D C R2 defined by D = {(x, y) ( R' [ x 2 0, y 2 0,ry 21} (a) Draw the set D. b) Draw the level curves of the function S(I, y) = (c) Using parts (a) and (b), argue whether the function f attains a maximum or a minimum value on D. (Note that you are not asked to compute these values) ry 2 1 (a) En el grafico el conjunto D es el que esta por encima de la curva de nivel zy = 1. Observemos que es la parte por encima del grafico de la funcion y = 1/2 con r > 0. (b) Las curvas de nivel de la funcion f estan representadas por las lineas rectas discontinuas, Observese que la curva de nivel dada for /(r, y) = c coincide con la curva de nivel de r + y = 1/c, que representa a la recta y = 1/c- r. (c) En la figura se ha representado el gradiente VS, que como se sabe apunta en la direccion en la que la funcion crece. Como se ve la funcion alcanza un maximo en el punto en el que la curva de nivel de f es tangente a la curva ry = 1, ya que se observa graficamente que en todas las demas curvas de nivel de f que intersectan a D, el valor de f es mas pequeno. Por otra parte, partiendo de cualquier punto de D y moviendonos en la direccion de -V / permanecemos dentro del conjunto D y la funcion f decrece estrictamente, por lo que no se alcanza ningun minimo en D. (6) Consider the function /(r, y) = -8ar - 2by' + cry + 5x - 3y + 2. (a) Discuss, according to the values of the parameters a, b and c, when is f strictly concave, knowing that ab = 1, a 20, 6 20. (b) Compute the values of a, b and e assuming that the Taylor polynomial of f of second order about the point (1,0) is -x' - 16y' + 5x - 3y + 2 (7) Given the function /(x, y) = -5x2 - 8y' - 2ry + 42r + 102y. (a) Find the critical points and classify them. (b) Find the largest open set in which f is strictly concave and strictly convex. (e) Find the absolute maxima and minima. (8) Consider the function /(x, y, 2) = x2 - y' + 112 and the set M = ((x, y, =) ( R3 : 23 + y" + = = 9). (a) Write the Lagrange equations satisfied by the extreme points of f on the set M. (b) Determine the extreme points of f on el set M.(1) The function f (r, y) = 3x2 + ev represents the profit of a firm that produces a units of good I and y units of good 2. (a) Find the gradient of f at the point (1, 0). (b) Last year the company produced 1 unit of good 1 and 0 units of good 2. This year the company is forced to produce (1 + Ar, Ay). Knowing that Ar and Ay are very small and that the company chooses them so that the profit increases the fastest, compute approximately Ar Dy (2) Given the linear mapping / : R4 - R3 f(x1, 12, 23, 14) = (21 + 22 + 23, 2x1 + 2x2 + 2ra, 21 + 22 + 3.ra + 14) (a) Find the matrix of f with respect to the canonical bases and the dimension of the kernel and the image. (b) Find a basis of the image of f and a basis of its kernel. (c) Find a system of linearly independent equations of the kernel and the image of f. (3) Given the following system of equations, sty+z=1 2r + ay + 2: = b arty+= = 1 (a) Discuss the system according to the different values of a and b. (b) Solve the system for the case a = 2, b = 2. (4) Consider the matrix 3 A = 0 0 0 -1 3 (a) Find the characteristic polynomial and the eigenvalues. (b) Show that the matrix is diagonalizable. (c) Find the corresponding diagonal form and the matrix change of basis