Sign of third order derivative

Question

We have a concave function $f(x,y)$, i.e. the Hessian matrix has non-positive elements. Can we show $\dfrac{\partial^3f}{\partial x \, \partial y^2} \le 0$?

Answer 1

The answer is no. Take $f(x, y) = - (x - y)^4$ for example. We have $\dfrac{\partial^3 f}{\partial x \partial y^2} = -24 (x-y)$ which could be positive or negative depending on values of $x$ and $y$.