For which values of $a$ the function $u(x,y)=x^2+x^4+axy+y^2$ is convex? concave

Question

Consider the function
$$
u(x,y)=x^2+x^4+axy+y^2
$$
For which values of $a$ the function $u(x,y)$ is convex? concave?

I know how to solve such kind of problems for a single variable function $f(x)$. There I would have to calculate a second derivative $f''_{xx}$ and find intervals where $f''_{xx}>0$ (convex) and $f''_{xx}<0$ (concave).

But what approach should I apply here? Is it $d^2u(x,y)$ instead of $f''_{xx}$ and checking if the matrix for $d^2u(x,y)$ is positive-definite (for convex) and negative-definite (for concave)?

Answer 1

For convexness, $d^2u(x,y)$ is positively semi-definite. Concaveness is opposite.