Consider the constraint $$S_1 = \{(x, y) \; |\; \sqrt{x} + \sqrt{y} = 1 \}$$ How to use Lagrange Multipliers, when the constraint surface has a boundary?
In this case, after the Lagrange multiplier method gives candidates for maxima/minima, we need to check the "boundary points" of $S_1$, namely, $(1,0)$ and $(0,1)$ to get the global max/min. I can see that these two are "boundary points" intuitively when I plot the curve.
However, instead if the constraint set be
$$S_2 = \{ (x, y) \; |\; x^2 + y^2 = 1\},$$ then in this question, one answer states that for this constraint set, there is no "boundary point". Constrained Extrema: How to find end points of multivariable functions for global extrema
The only difference I see is that pictorially, one is a closed curve, but the other is not.
However, I am unable to see what is the mathematical definition that will allow me to conclude that $S_1$ has boundary points $(0, 1)$ and $(1,0)$ and $S_2$ has none?
Q) What is the definition of "end point" or "boundary point" being used here that explains both $S_1$, $S_2$.