infimum and supremum of two argument function in particular set

Here is the task:

Find supremum and infimum of $\displaystyle f(x,y)=x^2+2xy+y^2$, if $\displaystyle (x,y)\in D, D=\{(x,y)\mid 6x^2+y^2-1=0\}$.

I have tried two ways:

1) searching for partial derivatives in order to find local extrema, but here is what happens:

$\displaystyle \frac{\partial f}{\partial x}=2x+2y,

\frac{\partial f}{\partial y}=2x+2y$

So, it means that there is no local extreme points at all? Or all $\displaystyle (x,y)\mid x=-y$ are extreme points?

2) from D follows that $\displaystyle y=\pm \sqrt(1-6x^2)$. Then I can get function $\displaystyle g(x)=f(x,\pm\sqrt(1-6x^2))$ and look for extreme points of g(x).

How do you think? Which way could fit? Or am I wrong in both of them?