Hessian determinant of a function

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I have a function with two variables ($f(x,y)$). Hessian determinant of the function is always equal to zero: $f_{xx}f_{yy}-f_{xy}f_{yx}=0 ~~\forall x,y $ and both $f_{xx}$ and $f_{yy}$ are always positive. What can I say about $f$? Is it convex?