(1) Consider the following system of linear equations with two parameters a, be R +1+2: 2ar + (3a - 1 )and + (50 - 2)= 2+ 2a 2ar + (3a - 1)y + (50 - 2+ 63)= = 20- b+ 2 (a) State the Rouchee-Frobenius Theorem. 0.5 points (b) Classify the above system according to the values of a and b. |1 point (c) Solve the above system for the values a = 1, b = 1/2. |0.5 points (2) Consider the function 1(2.1) = 47 (1,U) # (0,0) (, ") = (0,0) (a) Compute the partial derivatives 2(0,0) and (0,0) and the gradient of the function / at the point (0,0). 1 point (b) Compute the directional derivative of the function f according to the vector e = (1, 1) at the point p = (0. 0). Determine if the function f is differentiable at the point (0, 0). 1 point (3) Consider the function /(z, y) = ] - 2 + 323 + 3y. (a) Compute and classify the critical points (if any) of the function f in the set R?. |1 point (b) Find the largest open subset S C R where the function / is convex. Compute and classify the critical points (if any) of the function / in the set S. 1 point (4) Consider the function /(z, y) = 4 lay. (a) Compute the plane tangent to the graph of the function f at the point p = (1, 1,0). 1 point (b) Compute the Taylor polynomial of order 2 of the function f at the point p = (1, 1). |1 point (5) Let /(z, y. =) = x + = and consider the sphere with equation ?' + y' + ?' = 1. (n) Verify that the assumptions of Lagrange's Theorem hold. Write the Lagrange equations and obtain the solutions of those equations. |1 point (b) Determine the extreme points of f on the sphere. Determine if those points are local or global extreme points. Justify the answer. |1 point(1) Consider the following system of linear equations with a parameter a E R. ar+ + = b ar + ay + = = 0 stay+ as Please, answer the following questions. (a) Classify the system according to the values of a. 1 point (b) Solve the above system for the values a = b= -1. 1 point (2) Consider the function /(x, y) = 4x - y and the set A= ((z,V) ER' : 05x