## Maxima and Minima Without Calculus, Volume 6The purpose of this book is to put together in one place the basic elementary techniques for solving problems in maxima and minima other than the methods of calculus and linear programming. The emphasis is not on the individual problems, but on methods that solve large classes of problems. The many chapters of the book can be read independently, without references to what precedes or follows. Besides the many problems solved in the book, others are left to the reader to solve, with sketches of solutions given in the later pages. |

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### Contents

PREFACE | 1 |

CHAPTER TWO SIMPLE ALGEBRAIC RESULTS | 15 |

CHAPTER THREE ELEMENTARY GEOMETRIC QUESTIONS | 47 |

CHAPTER FOUR ISOPERIMETRIC RESULTS | 77 |

CHAPTER FIVE BASIC TRIGONOMETRIC INEQUALITIES | 92 |

CHAPTER SIX POLYGONS INSCRIBED AND CIRCUMSCRIBED | 111 |

CHAPTER SEVEN ELLIPSES | 126 |

CHAPTER EIGHT THE BEES AND THEIR HEXAGONS | 144 |

CHAPTER NINE FURTHER GEOMETRIC RESULTS | 156 |

CHAPTER TEN APPLIED AND MISCELLANEOUS PROBLEMS | 179 |

CHAPTER ELEVEN EUCLIDEAN THREESPACE | 211 |

CHAPTER TWELVE ISOPERIMETRIC RESULTS NOT ASSUMING EXISTENCE | 226 |

POSTSCRIPT ON CALCULUS | 239 |

293 | |

299 | |

### Common terms and phrases

ABCD algebra AM-GM inequality angles arc PQ argument arithmetic mean arithmetic-geometric means calculus Chapter circle of radius circle x2 circumscribed constant sum convex polygon convex region coordinates denote ellipse equality iff equation equilateral triangle example Find the maximum follows formula function geometric gives Hence holds illustrated in Figure inner parallel polygon inscribed triangle integer intersection isoperimetric quotient isoperimetric theorem jeep larger largest area largest value line segment located Mathematical maxima and minima maximize maximum area maximum value minimize minimum nonconvex parallelogram perpendicular positive numbers proof prove Ptolemy's theorem quadrilateral rectangle regular n-gon regular polygon result satisfying shortest distance shown in Figure sides of lengths smallest values solution solve sphere square straight line strict inequality subtended surface area tangent line tetrahedron Theorem 2.2a tile the plane tion tosses triangle ABC triangle PQR unit circle vertex vertices volume zero