Mathematics for economics:

(1) Consider the following system of linear equations with two parameters a, be R cty+ 2z 2ar + (3a - 1)y + (50 - 2)2 2 + 2a 2ar + (3a - 1)y + (5a - 2 +63)2 = 2a - 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\f(b) Compute the directional derivative of the function f according to the vector v = (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 f(x, y) = y - r + 3x3 + 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 f is convex. Compute and classify the critical points (if any) of the function f in the set S. 1 point(5) Let f(x, y, 2) = x + = and consider the sphere with equation a +y' + 2? = 1. (a) 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