Show that $f(x,y)=x^2+4y^2-4xy+2$ has an infinite amount of stationary points

Question

$f(x,y)=x^2+4y^2-4xy+2$

So, $f_x=2x-4y$ and $f_y=8y-4x$

To find the stationary points we have to equal the partial derivatives to $0$:

$2x-4y=0$

$8y-4x=0$

Because we cannot find an $x$ and $y$ via the system of equations, does that mean that there are a infinite amount of $x$ and $y$ that satisfy the equation, thus proving that there indeed are infinite stationary points?

Answer 1

Under the observation by Semiclassical: $$f(x,y)=(x-2y)^2+2,$$ you can see that that $f(x,y)\ge2$ since $(x-2y)^2\ge 0$, with $f(x,y)=2$ when $x-2y=0$. Then $x-2y=0$ is a critical line of minima.