Newbie question here. So I am starting to learn about constrained optimization in my multivariable calculus course and I was taught how to use the Lagrangian and Lagrange multipliers to solve an optimization problem subject to a specific constraint. I was also taught before this how to solve an optimization problem without using the Lagrangian by converting the objective function into a single variable one using the constraint equation and finding its critical point.
Now, when I did a problem subject to an equality constraint using the Lagrange multipliers, I succeeded to find the extrema. However, when I did the method where you restrict the objective function into a single variable using the constraint equation, there is no critical point and hence I could not solve. What seems to be the problem here?
This is the specific problem:
Let f(x,y) = (x^2) + (y^2) - 2y. Find the absolute extrema on the circle: (x^2) + (y^2) = 4.
Using the Lagrange multipliers, I found the absolute maximum at (0,-2) and absolute minimum at (0,2).
If I, instead, substitute the (x^2) term on my objective function as (4 - (y^2)), it becomes 4 - 2y. Getting the derivative with respect to y gives -2 and hence it has no critical points.