I'm trying to find the critical points for $f(x, y) = 5x^2-2y^2+10$ on the circle $x^2+y^2 = 1$.
I understand that when I substitute $x = \cos(t)$ and $y = \sin(t)$, I get all four critical points: $(0, 1), (0, -1), (1, 0)$, and $(-1, 0)$. However, if I solve for $x^2$ in the constraint, I get $x^2 = 1-y^2$, and when I plug this into the function, I get $f(y) = -7y^2 + 15$. $f'(y) = -14y$, so $f'(y) = 0$ when $y = 0$.
However, this only provides $2$ of the $4$ critical points, $(0, 1)$ and $(0, -1)$. You only get the other two critical points (when $y = 1$) when you do the similar substitution for $y^2 = 1-x^2$.
Why is this? It's usually sufficient to "merge" the equations once and get the derivative to find all critical points. Why is it necessary to do so with both of them in this case?