GreenZorg

Hello, I need to find a two-arguments function u(x,y) which satisfies six constraints on its derivatives. 1&2: On the first derivatives: du/dx>0 for all x & du/dy>0 for all y (so u is increasing in x and y) 3&4: On the second derivatives: d²u/dx²<0 for all x & d²u/dy²<0 for all y (so u is concave in x and y) 5&6: On the crossed derivatives: d²u/dxdy<0 for all x+y<theta (or at least y<theta) & d²u/dxdy>0 for all x+y>theta (or at least y>theta) (theta is a threshold) I found one specific function that satisfies those conditions: u(x,y)=xy+1-exp(theta-x-y) But I don't think this is the only one. I would like to find the most general function that satisfies those six conditions. The best would be that this specific function that I found, belong to a pretty well-known category of functions. Don't know if it is possible. Maybe Weibull functions? Did not try yet. Could you help me please? Thanks a lot GreenZorg sorry I forgot to say that x and y are quantities so they are positive. As a matter of fact, theta is positive too