Let the problem:

max/min f (x, y)

constrained to

x² + y² 1,

x² + y - 1 0

x + y -1.

where f is a differentiable and concave function on R (real numbers) and X is the set of feasible solutions of the problem.

(a) Reasonably explain whether the following assumptions are true or false:

If z X and f(z)=0, then f reaches a maximum in z with respect to X.

If zX and f(z)0, it is certain that z is not an optimal solution of f with respect to X.

The minimum of f in X can be reached at a point on the boundary that is not a vertex.

(b) Let f(x,y) = 2x + 2y - (x+y)²

Do points (1,0) and (0,1) satisfy the Kuhn-Tucker conditions? Are optimal solutions? Are they unique?

Does the point (0,1) satisfy the Kuhn-Tucker conditions? What can be concluded about this point?