I am trying to prove that the function define on $\mathbb R^2$ is convex. $$ f(x_1,x_2)=|x_1x_2|+x_1^2+x_2^2 $$ I know I should try to compute hessian, but the term $|x_1x_2|$ makes it hard to do so. Is there any trick to do this problem? I am still want to use hessian but not to use the basic definition, i.e., take two points and try to compute the linear combination of those two points...
Thank you!