These videos are well done, but the video above is a bit off on one technicality. Maximizing the Lagrangian with respect to the multiplier (at 11:57-12:20, and also mentioned briefly after 10:51) is not the best way to think about it because it gives wrong/nonsensical answers in some cases. A better way to think about it: the solution(s) must satisfy the constraints. This technicality will not be relevant in the course, but at the same time I don't want to have an error in the presentation without bringing it to your attention.