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I need to find atwo-arguments function u(x,y)which satisfiessix constraintson its derivatives. x and y are quantities so always positive.

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<θ (or at least y<θ) & d²u/dxdy>0 for all x+y>θ (or at least y>θ) (θ is a threshold)

I found one specific function that satisfies those conditions: u(x,y)=xy+1-exp(θ-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

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# Find a 2-arguments function from six constraints on its derivatives

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