Partial and Cross Derivative Relationship

Question

Suppose I have a function $F(x,y)$. I know that $\frac{dF}{dx}<0$ and $\frac{dF}{dy}<0$.

Can I conclude that $\frac{d^2F}{dxy}\leq0$?

If not under what conditions would it be possible?

Generally is there any relationship between each partial derivative and the cross-derivative?

Answer 1

Let F(x,y)=-x-y+xy.

Then dF/dx=-1+y and dF/dy=-1+x while d2F/dxdy = 1.

Thus, at least at the origin (0,0) and a small neighborhood including the origin, the given hypotheses do not imply d2F/dxdy is less than or equal to zero.

Answer 2

You're trying to make a claim about the sign of a second derivative based on the values of first derivatives. This sort of idea doesn't work out in practice.

For your particular question, consider $F(x, y) = xy$. At the point $(-1, -1)$, $\frac{\partial F}{\partial x} = y < 0$ and similarly $\frac{\partial F}{\partial y} < 0$. However, $\frac{\partial^2 F}{\partial x \partial y} = 1 > 0$.