Application of Cauchy-Schwarz inequality method for resolving constrained optimization problems at classroom with formal pre-operational phase of thinking

The optimization problem is a mathematical problem that is introduced at the junior high school level. Material related to optimization has even been introduced for elementary school levels in students with a level of concrete operations, especially for students who are prepared to take part in the Math Olympiad. The problem arises here, is that complicated methods for finding solutions to optimization problems, such as the simplex method, ordinary differential equations, partial differential equations, and the Lagrange multiplier were only introduced in high school or in college. To bridge this we need a special method that is easily understood by students participating in the pre-Olympic who are at the level of formal preoperative cognition, one of these methods is called the Cauchy-Schwarz inequality method. The Cauchy-Schwarz inequality is a powerful tool for finding the maximum and minimum solutions for a target equation with one or two constraint equations. The interesting thing is that the prices of the optimum manufacturing variables are reached when the Cauchy-Schwarz equation is fulfilled.