So we have the problem: maximize $x^2+y^2$ subject to $x^2-y \leq3$ and $y\leq 1$. And I sorted out the KKT conditions for the problem (is here where the problem is?):
$2x=\lambda _12x$,
$2y=\lambda_2-\lambda_1$,
$\lambda_1\geq0$ with $\lambda_1=0$ if $x^2-y<3$,
$\lambda_2\geq0$ with $\lambda_2=0$ if $y<1$.
So I check the KKT condition one by one: both constraints are active, only one is active, none is active, etc.
But my problem comes as follows: suppose the first constraint is active and the second inactive. So we have $y<1$, $\lambda_2=0$, $2y=-\lambda_1$.
I don't know what to proceed after that, I mean, say if $x=0$, then $\lambda_1$ can take any value. So what does it mean when $\lambda$ is supposed to be a constant?