2-arguments function derived from 6 conditions

[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