I want to find the maximum of $f(x,y) = x^2 - y^2$ under the constraint $\frac12 x^2 + y^2 - 1 = 0$.
I defined Lagrange function:
$$ L= x^2 - y^2 + \lambda \left( \frac12 x^2 + y^2 - 1 \right) $$
Then caculated $L'_x, L'_y, L'_{\lambda}$ and got the following equations system:
$$ \begin{aligned} x (2 + \lambda) &= 0 \\ 2 y (\lambda - 1) &= 0 \\ \frac12 x^2 + y^2 - 1 &= 0 \end{aligned} $$
How to solve this system so I can find the maximizer $(x,y)$?
($\lambda = 1$, $x=0$ and $y=\pm 1$) Or ($\lambda = -2$, $y=0$ and $x=\pm \sqrt 2$)