Let $f(x_1,x_2)=x_1x_2$, with $x_1,x_2 \ge 0$. Is $f$ a convex function? Why?
Edit (in view of the comments below)
The Hessian matrix is $H=[0\, 1; \,1 \,0]$, which is indefinite (in general). In fact, $x H x'= 2x_1x_2$. However, $2x_1x_2\ge 0$ when $x_1,x_2\ge 0$, so that the Hessian is indeed positive semidefinite when $x_1,x_2\ge 0$. Therefore, is this sufficient to conclude that my $f$ is convex?