Is the function $f(x,y) = xy$ (quasi) concave or (quasi) convex?
I start by finding the Hessian matrix, which is $H = \begin{bmatrix} 0 &1 \\ 1 &0 \end{bmatrix}$.
Then I ask myself if the matrix is negative semi-definite or positive semi-definite.
We have $x'Hx = \begin{bmatrix} x_{1}&x_{2} \end{bmatrix}\begin{bmatrix} 0 &1 \\ 1 &0 \end{bmatrix}\begin{bmatrix} x_{1} &x_{2} \end{bmatrix} = 2 x_{1} x_{2} $
Since in general $2 x_{1} x_{2}$ can have any sign, we cannot say whether the function is (quasi)convex or (quasi)concave.
Is this correct?