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Concavity of a function along a path

Suppose that $f(x,y)$ is a continuously differentiable function and $g(x,y) =xy-f(x,y)$. I know that $g$ is concave if and only if $(-f_{xx})(-f_{yy}) -(1-f_{xy}) ^{2}>0$ and $f_{xx}>0$. Now suppose …
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