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Maxima and Minima Without Calculus

Ivan Niven
Mathematical Association of America
Publication Date: 
Number of Pages: 
Dolciani Mathematical Expositions 6
BLL Rating: 

The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Kara Shane Colley
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Maxima and Minima without Calculus consolidates "the principal elementary methods for solving problems in maxima and minima."  Niven follows the rule:  "if a problem can be solved more simply by calculus, leave it to calculus."  Most of the extremal problems he discusses are algebraic or geometric in nature; they are often solved using the arithmetic mean-geometric mean inequality.  Niven explores isoperimetric problems for triangles, quadrilaterals, and inscribed and circumscribed polygons, as well as varrious other topics, such as trigonometry, ellipses, and Euclidean 3-space.  What is most striking about this book is that Niven shows how a proof that assumes a solution exists is incomplete because it doesn´t prove that a maximal or minimal solution must exist.  In the final chapter, Niven takes the reader through proofs that don´t assume existence. 

Niven intends Maxima and Minima to serve as a "resource book, not a textbook" because there are some problems left for the reader to solve, but not that many.  Professors of classes that deal with extremal problems, such as calculus, linear programming, and game theory, could enhance the teaching of their courses by including some of these problems.  Even so, undergraduate math majors could comprehend the text on their own.  Niven writes in a clear, understandable way and builds slowly up to the difficult material.  For instance, he uses concrete examples with numbers or specific functions before moving on to more generalized, abstract material.  Another benefit to students is that Niven provides solutions to all problems, including proofs.  Proofs are frequently the most difficult problems for students and all too often, textbook writers omit their solutions. 

Niven has a way of bringing this topic to life.  "Maxima and Minima" is not a dry, boring math text.  There is a person behind the mathematics, making the material enjoyable.  Throughout the book, Niven sprinkles short anecdotes and novelties within his prose.  For example, he offers a mnemonic device for remembering the decimal expansion of e:

 He  studied  a  treatise  on  calculus
 2.  7  1  8  2  8

The best part of the book are the problems he chooses.  There are legends (the problem of Dido), famous problems (Fermat´s problem, also known as the airport problem), and real life problems (tacking a sailboat against a headwind).  These problems are more exciting to solve than problems without a context.  When I was a student, I loved it when my professors introduced famous problems because it made me feel "well-read" in the field of mathematics.  Above all, Niven´s problems are interesting and his explanations are clear.  This is the highest (maximum)! praise I can give a mathematics text. 

Kara Shane Colley studied physics at Dartmouth College and math education at Teachers College. She has taught math and physics to middle school, high school, and community college students in the U.S., the Marshall Islands, and England. Currently, she is travelling around Mexico, volunteering on organic farms and learning Spanish. Contact her at


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