What we say about the convexity, quasiconvexity, concavity and quasiconcavity of this function?
The Hessian is $$\begin{bmatrix} y &1 \\ 1& x \end{bmatrix}$$ We can't say anything about the sign unless we know the values of x and y.
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Updated. The hessian does not help – onelessproblem May 15 '17 at 03:24
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1Related: See here – Chinnapparaj R May 15 '17 at 03:26
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1Your Hessian is not right, recheck the calculations. – dxiv May 15 '17 at 03:29
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What is the domain of the function? How about checking on the line joining the two points $(1,0)$, $(0,1)$. – copper.hat May 15 '17 at 04:04