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.

Contents:

• About the Authors
• Introduction to Part I:
  • Considering Extreme Cases
• Introduction to Part II:
  • Finding Optimal Values by Using the Square Inequality — The Vertex of a Parabola
  • Finding Optimal Values by Using the Triangle Inequality
  • Finding Optimal Values by Using the Aspects Product of Factors with Constant Sum and Sum of Numbers (Values, Terms) with Constant Product
  • Generalization of Chapter 4 to More Than Two Factors/Values
  • More Unusual Kinds of Optimum Considerations
  • Maximum Triangles in Parallelograms — Maximum Parallelograms in Triangles
• Conclusion
• Appendix
• Index

Readership: This book is an ideal resource for mathematics teachers and students at the secondary school level. It may also appeal to the general readership interested in mathematics.