$f(x) = \begin{cases} e^{x_1x_2} & x_1, x_2 \geq 0 \\ +\infty & otherwise \\ \end{cases}$
The Hessian of this matrix is: $H = \begin{pmatrix} x_2^2e^{x_1x_2} & x_1x_2e^{x_1x_2} \\ x_1x_2e^{x_1x_2} & x_1^2e^{x_1x_2} \end{pmatrix}$
Using sylvestor's criterion to examine positive semi-definiteness;
$\checkmark$ $x_2^2e^{x_1x_2} \geq 0$
$\checkmark (x_1x_2)^2(e^{x_1x_2}) - (x_1x_2)^2(e^{x_1x_2}) = 0$
Therefore the matrix is positive semidefinite, is it not?
The suggested solutions indicate the function is convex, and hence the hessian is not positive semidefinite.