Think of a function $R^2\rightarrow R$ satisfies that there exists some point where each partial derivative is positive and mixed partial derivative is negative?
$$f_x>0,f_y>0,f_{xy}<0$$
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Did you want this to hold everywhere or at a single point? – copper.hat Oct 06 '16 at 01:13
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Consider the function \begin{align} f(x,y)=-xy \end{align} at the point $(-1, -1)$. Observe \begin{align} f_x(x, y) = -y \ \ \text{ and } \ \ f_y(x, y) = -x \end{align} and \begin{align} f_{xy}(x, y)=-1. \end{align}
Edit: Here's the example for everywhere on $\mathbb{R}^2$. Let us consider the function \begin{align} f(x,y) = -e^{-x-y}. \end{align} Observe \begin{align} f_x(x,y) = e^{-x-y} \ \ \text{ and } \ \ f_y(x, y)= e^{-x-y} \end{align} and \begin{align} f_{xy} (x, y) = -e^{-x-y}. \end{align}
Jacky Chong
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2I wonder if there's a function that satisfies these properties everywhere? – Thomas Rasberry Oct 06 '16 at 01:21
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Yup! I was running configurations of e^f(x,y) on Wolfram starting with e^(x+y) and stumbled upon it. Nice! – Thomas Rasberry Oct 06 '16 at 01:38