Maximum and minimum of a function of 2 variables (Lagrange multipliers)

My question is : **Can I solve the problem without passing by Lagrange multipliers?** Because I have some difficulties by doing so. **If not, then I don't want any tip and I'll work harder.**

I must find the minimum and the maximum of the function $\displaystyle f(x,y)=x^2+y^2+\frac{2xy\sqrt{2}}{3}$ in the ellipse $\displaystyle x^2+2y^2 \leq 1$.

So it seems a problem that can be solved via Lagrange multipliers, but I didn't find a way to find $\displaystyle \lambda$, $\displaystyle x$ and $\displaystyle y$. (I even had to divide by $\displaystyle 0$ if $\displaystyle \lambda \neq 1$ but then I found another condition ( if $\displaystyle \lambda \neq 0$ then ...) and even with it I couldn't isolate $\displaystyle \lambda$ nor $\displaystyle x$ nor $\displaystyle y$).