Let the problem min/maxf(x,y) subject to:

• x2+y21,
• x2+y10,
• x+y1 .

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

a) Explain if the following assumptions are true or false:

• If zX and f(z)=0 , then f reaches a maximum respect to X.
• If zX and f(z)=0 , is sure that z is not an optimal solution of f 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?