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The inequality of arithmetic and geometric means from multiple perspectives. (English)
Math. Teach. (Reston) 109, No. 4, 314-318 (2015).
Summary: NCTM’s Connections Standard recommends that students in grades 9‒12 “develop an increased capacity to link mathematical ideas and a deeper understanding of how more than one approach to the same problem can lead to equivalent results, even though the approaches might look quite different” [{\it NCTM}, Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics (NCTM) (2000), p. 354]. In this article, the authors embody these recommendations by exploring the AGM inequality in three variables. First, they give an algebraic proof of the AGM inequality using the factorization of a certain symmetric polynomial. Next, they describe geometric inequalities called isoperimetric inequalities and explain how they are related to the AGM inequality. Along the way, they give an elegant proof of the AGM inequality by Cauchy and introduce a beautiful extension called Maclaurin’s inequalities. Finally, the authors give yet another proof of the AGM inequality using Rolle’s theorem and calculus. By studying the AGM inequality in three variables and from these different perspectives, both teachers and students can experience and gain appreciation for the interconnected nature of mathematics. (ERIC)
Classification: H30 I10