Finding local and global extrema even when the determinant of the Hessian zero

Finding local and global extrema even when the determinant of the Hessian
zero

I am trying to solve the following problem:
Let $f: \mathbb R^2\rightarrow\mathbb R$ be a function defined by $$
f(x,y) = x^{2n} + y^{2n} - nx^2 + 2nxy - ny^2, $$ where $n$ is a natural
number greater than 1. Decide whether $f$ has a (global) minimum. Also
find all the points at which $f$ attains its local maxima and local
minima.
I calculated the partial derivatives $f_x = 2nx^{2n-1}-2nx+2ny$, $f_{xx} =
2n(2n-1)x^{2n-2}-2n$, $f_{xy} = 2n$, and so on. But I got an equation
system $f_x = f_y = 0$ of degree three, which I had hard times solving it.
I found out that $(x,y) = (0,0)$ is one of its solutions, but at that
point the the determinant of the Hessian is zero, from which I could not
conclude whether it was a local extremum.
I was not sure about how to prove that the minimum value of $f$ exists,
either.
I would be most grateful if you could help me solve this problem.,