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
