By Titu Andreescu
Questions of maxima and minima have nice useful value, with functions to physics, engineering, and economics; they've got additionally given upward push to theoretical advances, particularly in calculus and optimization. certainly, whereas such a lot texts view the examine of extrema in the context of calculus, this conscientiously developed challenge e-book takes a uniquely intuitive method of the topic: it offers enormous quantities of extreme-value difficulties, examples, and strategies essentially via Euclidean geometry.
Key positive aspects and topics:
* entire choice of difficulties, together with Greek geometry and optics, Newtonian mechanics, isoperimetric difficulties, and lately solved difficulties equivalent to Malfatti’s challenge
* Unified method of the topic, with emphasis on geometric, algebraic, analytic, and combinatorial reasoning
* Presentation and alertness of classical inequalities, together with Cauchy--Schwarz and Minkowski’s Inequality; easy ends up in calculus, equivalent to the Intermediate price Theorem; and emphasis on basic yet necessary geometric innovations, together with variations, convexity, and symmetry
* transparent suggestions to the issues, frequently followed by means of figures
* thousands of routines of various hassle, from basic to Olympiad-caliber
Written by way of a staff of validated mathematicians and professors, this paintings attracts at the authors’ adventure within the school room and as Olympiad coaches. by means of exposing readers to a wealth of inventive problem-solving methods, the textual content communicates not just geometry but in addition algebra, calculus, and topology. excellent to be used on the junior and senior undergraduate point, in addition to in enrichment courses and Olympiad education for complicated highschool scholars, this book’s breadth and intensity will attract a large viewers, from secondary university lecturers and students to graduate scholars, specialist mathematicians, and puzzle enthusiasts.
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Additional resources for Geometric Problems on Maxima and Minima
24 A right circular cone of volume V1 and surface area S1 and a circular cylinder of volume V2 and surface area S2 are circumscribed about the same sphere. Prove that: (a) 3V1 ≥ 4V2 ; √ (b) 4S1 ≥ (3 + 2 2)S2 . 25 Two balls are given in space with no common points. Find the position of a light source on the line connecting the centers of the balls such that the lighted part of the boundary spheres has a maximal total area. 4 The Method of Partial Variation The method of partial variation uses the simple observation that if a function of several variables has a maximum (or a minimum) with respect to all variables, then it also has a maximum (or a minimum) with respect to any subset of variables.
Not lying in a plane) edges of a given cube. Find the position of these three points such that the perimeter of triangle X Y Z is a minimum. 2. Employing Algebraic Inequalities 23 Solution. Assume that the given cube ABC D A1 B1 C1 D1 has edge of length 1. Without loss of generality we will assume that X lies on the line determined by C1 D1 , Y on the line AD and Z on the line B B1 (Fig. 16). Figure 16. Consider the coordinate system in space with origin A and coordinates axes AB, AD, and A A1 .
Intermediate Value Theorem. If f (t) is continuous in the ﬁnite closed interval [a, b] and f (a) · f (b) < 0, then there exists at least one t ∈ (a, b) such that f (t) = 0. We start with an example in which a quadratic function is used. 1 Two ships travel along given directions with constant speeds. m. the distance between them is 20 miles, at 9:35 the distance is 15 miles, while at 9:55 the distances is 13 miles. Find the time when the distance between the ships is a minimum. Solution. Assume that one of the ships travels along a line g, while the second travels along a line h.
Geometric Problems on Maxima and Minima by Titu Andreescu