Finding Minimum Value Using Cauchy-Schwarz Inequality

This GeoGebra applet demonstrates how to use the Cauchy-Schwarz inequality to find the minimum value of an algebraic expression under a given constraint. The problem involves three positive real numbers a, b, and c that satisfy the condition a + b + c = 3. The goal is to find the minimum value of the expression 1/a + 4/b + 9/c. The applet walks students through the elegant method of constructing two vectors: the first vector has components (sqrt(a), sqrt(b), sqrt(c)), and the second vector has components (1/sqrt(a), 2/sqrt(b), 3/sqrt(c)). By applying the Cauchy-Schwarz inequality, which states that the dot product of two vectors is less than or equal to the product of their magnitudes, the algebraic expression is transformed into a clean vector inequality. This allows students to see how the constraint a + b + c = 3 directly leads to the minimum value of the target expression. The interactive interface lets students manipulate the values of a, b, and c while keeping their sum equal to 3, observing in real time how the expression 1/a + 4/b + 9/c changes and converging toward its minimum. Students can visually confirm that the minimum occurs when a = 1/2, b = 1, and c = 3/2, yielding a minimum value of 9. The geometric interpretation via vectors reinforces the connection between algebra and geometry, helping learners internalize the Cauchy-Schwarz inequality beyond rote memorization. This applet is designed for high school mathematics students studying inequalities and optimization. It serves as an excellent supplement to classroom instruction on the Cauchy-Schwarz inequality, offering both visual intuition and algebraic rigor. By allowing students to explore the relationship between vector operations and algebraic minimization, the applet deepens conceptual understanding and builds problem-solving confidence for competition-style questions and advanced coursework.