What can we say about the function which has following kind of Hessian matrix?

Question

I have function $f(x,y)$ with $x\geq0$ and $y\geq0$. The Hessian matrix of this function has following properties.

1- $f_{xx}>0$

2- Determinant of the Hessian matrix is zero.

Can we say that the function $f(x,y)$ is jointly convex over $(x,y)$ for all of the desired domain?

Answer 1

Yes. The Hessian of $f$ is semi positive definite every on the domain , because it passes the Sylvester's criterion .

So $f$ is jointly convex on its domain.