(1) Consider the set A = ((r.y) ER' :x'ty' 52, y21'). (a) Draw set A, its boundary and interior set. Discuss whether A is open, closed, bounded, compact and /c convex. Clearly explain your answer. (b] Consider the function f(r.y) = if (x, y) # (0,1). if (r, y) = (0, 1). Determine If this function achieves a maximum in A. Does it achieve a minimum? (2) Consider the following system of equations (a) Show that the above system of equations determines y and = as differentiable functions of r in a neighborhood of the point (3, 1,2). (b) Let y(x), =(x) be the functions found in part (a). Compute the derivatives y'(3) y ='(3). (3) Consider the function f(ry) = x - y' - 2ry -x3 (a) Determine the greatest open and convex set S of R" where the function f is concave. (b) Determine whether f achieves global extrema over the set S that you identified in the answer to the previous question. (4) Consider the function f(z, y) = >+y' -2ry and the set A = ((r, y) e R' : x' + y' = 2) (a) Write the Lagrange's equations that are needed to identify the extrema of f on A. (b) Determine the global extrema of f on A, and clearly argue whether they are a minimum or a maximum. (5) Consider the following maximization problem MAX rtyty-1 sa. r+