I have recently learned about the method of Lagrange multipliers in calc 3 and I understand it for the simple cases presented in lecture such as this one, where the constraint g is a circle and the objective f has roughly parallel level curves that are monotonically increasing as we move to the right
However, what if I were to change the objective function f such that the w = 12 level curve was instead the w = 20 level curve? Now the local max of f on the constraint circle would occur on a level curve that cuts through the constraint circle, rather than one tangent to it. It was my understanding that there would always be tangency with the level curve and constraint at the local max/min on the constraint. Does the Lagrange multipliers method still work in this case? If not, why not (what condition have I violated)?