Solve the questions below;

[1} Consider the following system of linear equations :r. 22 :cas :c+[2b}y whereo,bER. {a} Classify the system acmrding to the values of s and i}. {h} Solve the above system for the 1values of o and i:- for which the system has infinitely many solutions. [2} Consider the function II II H H' 2 new = {39E :5 (my: a {W '1 5i {1'1 3"] = [0,\"). {:1} Determine whether the function f is continuous at the point {0,0}. {h} Compute [if they exist] the partial derivatives of f at the point [0,0]. Compute [if it exists} the derivativeof f at the point [i], ll} according to the vector u = {1,4}. gIs the function f differentiable at the point {0.0}? [3} Consider the function fling y} = 411:2 + [o + my\" + Easy + 2. with u.b e n {a} Study the concavity and the convexity of the function f according to the values of a. and fl. {h} For the values o. = 1., b = i], does the function f attain a maximum andfor minimum value in R2? At what points? {Justify the answer]. [4} Consider the function I (31 y} = 37y: and theset A: {fey} Elli!2 : 3+3; 5 lyz+y rm}. {a} Compute the Kuhn-Tucker equations that determine the extreme points of f in A. {h} Compute the solutions of the above Kuhn-Tucker equations. [5} Consider the function f {31 y} = I: + 9'2 andthesetA={[r,y} ER3::r2y+=ll}. {a} Compute the Lagrange equations that determine the extreme points of f in A and find their solutions. {h} Using the second order conditions. characterise the above solutions of the Lagrange equations as corresponding to maximum or minimum values. Can you justify if any of those points corresponds to a global maximum or minimum? [Justify the answer}