Help to construct a function that satisfies the following conditions

Question

I am struggling to come up with a function $f(x_1,x_2)$ that satisfies the following conditions

• $f(x_1,x_2) \in (0,1)$
• $f(x_1,x_2)$ differentiable both in $x_1$ and $x_2$
• $f(x_1,x_2)$ strictly increasing in both $x_1$ and $x_2$
• $f(x_1,x_2)$ concave in both $x_1$ and $x_2$
• $\frac{\partial^2 f(x_1,x_2)}{\partial x_1 \partial x_2} > 0$

I can think of a simple example like $f(x_1,x_2) = x_1 \cdot x_2$, with $x_1 \in (0,1)$ and $x_2 \in (0,1)$. But I struggle with thinking of alternative functional forms.

Would anyone care sharing some thoughts? Thank you all for the wonderful help!

Answer 1

What about $f(x,y)=\sqrt(x)+\sqrt(y)+xy$.

Then it satisfies the fifth point.

It is concave by viewing its graph. Other can verified easily.