Maxima and Minima

This book explores the versatile concept of maxima and minima - extremes and optimal values - across a broad spectrum of mathematics. Far from being limited to calculus, the concept of extremes is applicable to nearly every field of mathematics, including geometry, arithmetic, algebra, and stochastics, offering powerful tools for problem solving. The first part of the book introduces a wide range of applications of the method of extremes, showing how it can be used to tackle problems both in everyday life and in mathematical contexts. The second, larger part focuses on solving extreme value problems without relying on derivatives, instead building reasoning and proof from first principles. Here, special emphasis is placed on geometry, where questions of optimality often reveal surprising insights about shapes and figures. By approaching mathematics through the lens of extremes, readers will gain a deeper understanding of core concepts and discover fresh ways to enrich both learning and teaching.

This book explores the versatile concept of maxima and minima - extremes and optimal values - across a broad spectrum of mathematics. Far from being limited to calculus, the concept of extremes is applicable to nearly every field of mathematics, including geometry, arithmetic, algebra, and stochastics, offering powerful tools for problem solving. The first part of the book introduces a wide range of applications of the method of extremes, showing how it can be used to tackle problems both in everyday life and in mathematical contexts. The second, larger part focuses on solving extreme value problems without relying on derivatives, instead building reasoning and proof from first principles. Here, special emphasis is placed on geometry, where questions of optimality often reveal surprising insights about shapes and figures. By approaching mathematics through the lens of extremes, readers will gain a deeper understanding of core concepts and discover fresh ways to enrich both learning and teaching.