Partial and Cross Derivative Relationship

1,611

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.

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Updated on October 04, 2020

Comments

  • Admin
    Admin about 3 years

    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?