Concluding critical points of function are not in the domain for $f(x,y) = x^{3} - x +y^2 - 2y$ My question has to do more with determining if the points are in the domain or not. So I am asked to find the extrema of the function $f(x,y) = x^{3} - x +y^2 - 2y$ over the closed trinagular region $(-1,0), (1,0), (0,2)$.
Taking the necessary derivatives I find the following "critical points"
$$\left(\pm \sqrt{\frac{1}{3}},1\right)$$
Now a solution I have says "it is easy to conclude that the two points are not in the domain". My question is how is this conclusion drawn? 
One way I wanted to know if it was correct was if I took the equation of the line between $(-1,0)$ and $(0,2)$ which is $y = 2x + 2$. If I plug my $x$ value from my critical point I don't get the corresponding $y$ value (in this case 1). Would that be the way to verify the points are not in the domain or is there another method to go about things?

What you have is fine.
We can say that the triangle is bounded by the lines
$y = 2x + 2, y = -2x + 2, x = 0$
or the region is below the line 
$y = \begin {cases} 2x + 2 & x\le 0\\ -2x + 2 & x>0\end{cases}$
Plugging the points $x = \pm \sqrt {\frac 13} \approx \pm\frac {4}{7}$
In fact $\frac {4}{7}$ is sligthly less than $\sqrt {\frac 13}$ 
$y(\sqrt {\frac 13}) < y(\frac 47) = \frac 67   < 1 $
The line is below the point $(\sqrt {\frac 13}, 1)$
Once you have found that there are no critical points inside the boundary, we must assume that the extrema lie on the boundary.
