Let the problem max/minf(x,y) constraited to:

• x2+y21,
• x2+y10.
• x+y1.

where f is a differentiable and concave function on R2 and X is the set of feasible solutions of the problem.

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

• If zX 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?