The square form is $H:=x^T\nabla^2 f(x) x= 2 a b$ where $x=[a,b]$. Now $f(x_1,x_2)=x_1x_2$ in $\mathbb R^2_{++}$ (problem b).
I am perplexed:
I think my teacher means that this not positively semidefinite because $H>0$ -condition is not satisfied when $x\in \mathbb R^2$.
I think the question restrict the domain to $\mathbb R^2_{++}$ so $H>0$ so positively semidefinite.
my teacher says that there is only one definition for positively-semi-definiteness and they match.
By different domains with each pos.semi-definiteness -definition, I got different answers so not matching definitions. I think I tried the the determinant rule -thing.
Now is this function positively-definite and when? What is called definiteness when you restrict the domain? I feel it quite stupid if definites is really limited to $\mathbb R^2$ or $\mathbb C^2$.
Question B (source here)
Answer the question B (here)