Let f(x,y) be a smooth function.
If i want to find the min and max of this function in the quarter disk constrained by x^2+y^2=1 in the first quadrant.
Can i then use lagrange multiplier to do this or is the best way to go to just search for stationary points and then check the boundery?
Lagrange multipliers will work for "most" of the boundary if you take the boundary to be 3 pieces: $x = 0$, $y = 0$, $x^2 + y^2 = 1$. But then you need to check the points $(1,0)$ and $(0,1)$ and $(0,0)$ separately because the min or max could happen at one of these points and you might not find it with Lagrange multipliers on the 3 boundaries.