Novel insights into the multiplier rule

We present the Lagrange multiplier rule, one of the basic optimization methods, in a new way. Novel features include: • Explanation of the true source of the power of the rule: reversal of tasks, but not the use of multipliers. • A natural proof based on a simple picture, but not the usual technical derivation from the implicit function theorem. • A practical method to avoid the cumbersome second order conditions. • Applications from various areas of mathematics, physics, economics. • Some hnts on the use of the rule.