Lagrange Multiplier Method Introduced from View of Geometry

The Lagrange multiplier method is derived from the view of geometry. Furthermore, the solutions given by the Lagrange multiplier method are not necessarily minimal solutions about the conditional extremum problem. A sufficient condition of second order is given for that solutions given by the Lagrange multiplier method are minimal solutions of the conditional extremum problem. When this second order sufficient condition is used for a judgment method, it is more convenient than other judgment methods.