The Lagrangian is:
$L(x,\lambda) = x_1x_2-2x_1-\lambda (x_1^2-x_2^2)$
Taking the derivatives and setting it equal to zero gives:
$x_2-2\lambda x_1-2=0$
$x_1+2\lambda x_2=0$
$x_1^2-x_2^2=0$
The points satisfying the conditions above are found to be: $(1,1)$ with $\lambda = -\frac{1}{2}$ and $(-1,1)$ with $\lambda = \frac{1}{2}$
This is the point where I get lost, how can I found out which one of them is a maximizer or minimizer? I usually used the Hessian to find the eigenvalues to see if it is greater or less than zero.
But the Hessian for the first point becomes
$\begin{bmatrix} -2\lambda & 1 \\ 1&2\lambda \end{bmatrix}$= $\begin{bmatrix} 1 & 1 \\ 1&-1 \end{bmatrix}$ which has eigenvalues
$\lambda_1 =1.41$ and $\lambda_1 = - 1.41$ but this is not correct. I get the same eigenvalues for the other point. I really need someone's help !