140 CHAPTER 3 cost minimization problem with equality constraints as a "carrot and stick" policy to force constraints, without the use of force. The idea is to let prices do the work. 3.2 MAIN RESULT: LAGRANGE MULTIPLIER RULE 3.2.1 Precise formulation of the multiplier rule Our goal is to express the principle of Lagrange in a precise way. Let x be a point of R n , > 0, and let f i : U n (x, ) R, 0 i m, be functions of n variables, defined on an open ball with center x, such that f i (x) = 0, 1 i m. These functions might be defined outside the ball as well, but this is not relevant for the present purpose. Definition 3.1 Main problem: equality constrained optimiza- tion. The problem f 0 (x) extr, f i (x) = 0, 1 i m, (P 3.1 ) is called a finite-dimensional problem with equality constraints. The following two examples illustrate this type of problem, and the possibility of solving such problems without Lagrange's method. Example 3.2.1 Solve, for a given constant a > 0, the problem f 0 (x) = x 1 x 2 max, f 1 (x) = x 2 + x 2 - a 2 = 0, x i > 0, i = 1, 2. 1 2 Solution. · Below, we will solve this problem with the method of the La- grange multiplier rule. · Here we note that the problem has a trivial geometrical solution (Fig. 3.1), using that the area of the shaded triangle equals 1 1 ah = x 1 x 2 , 2 2 which reduces the problem to that of maximizing the height h. · Alternatively, we can easily reduce to an unconstrained problem x 1 a 2 - x 2 max, 0 < x 1 < a, 1 and solve this using the Fermat theorem.