I understand that in general, the KKT conditions are not sufficient for optimality. However, if the primal problem is a convex optimization problem, then the KKT conditions are sufficient for optimality (given smoothness of functions).

I'm trying to understand how this relates to the method of Lagrange multipliers that is typically taught in Calculus. Wikipedia doesn't seem to mention anything about convexity in regards to this method. I thought the KKT conditions were supposed to generalize this method, but maybe I am confused. Can someone explain the relationship?