Consider the function $f(x,y) = x^2 + xy + y^2$ defined on the unit disc $D = \{(x,y) \mid x^2 + y^2 \leq 1\}$.
I can not simplify the equations to the point where I find a constant for the lagrange multiplier and thus can't find the points of the extrema. I used the method that we can create a new function $L$ with the variables $x$, $y$ and $\lambda$ where $\lambda$ is the Lagrange multiplier. Then found the partials of $L$ with respect to each variable. This is where I am stuck because I can not simplify enough to find $x$ and $y$. Are there any tricks for this type of question?