I am given the function $F(x,y) = 2x^2+y^2-y$ and the boundary $x^2+y^2≤1$. After finding the critical points inside the boundary, to find the critical points on the boundary, I substituted $1-y^2$ in for $x^2$ into the equation, found the derivative, set it equal to 0, and found the critical points on the boundary.
My professor told me I couldn't do it that way and had to parametrize the function in terms of t in order to find the critical points on the boundary. Is he right? and why? I've seen both methods work online and I got the same answer as him.