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# Class Note for MATH 294A with Professor Savitt at UA

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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 15 views.

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Date Created: 02/06/15
Inequalitiesquot Math 294A Problem Solving Seminar 7 Vera Furst March 217 2007 o A thmeticmeanigeometricmezm inequality For 11 In gt 07 1112 znwn S n with equality if and only if 11 12 In 0 CauchySchwarz inequality For all real a1 an and b1 bn n n 12 n 12 Zam 2a 2b 7 i1 i1 i1 with equality if and only if albl agbg anbn 2 0 ln vector form 512 S Hillllblly with equality if and only if E d for some nonnegative scalar 0 Example 1 Bernoullils inequality Prove that for 0 lt a lt 17 l I S l am for all z 2 71 How would the inequality change if a gt 1 or a lt 0 Example 2 Suppose that f R A R is a twice differentiable function with f z 2 0 for all z Prove that for all a lt I f i5 lt fa fb 2 7 2 Deduce the arithmeticmeaniquaclraticmean inequality for two numbers Example 3 If a b c d are positive numbers such that c2 d2 a2 b237 prove that a 12 7gt1 d7 with equality if and only if ad be Problem 1 Let a b c be the sides of a triangle Show that abbcca a2b2c2 2abbcca These problems or some version of them all appear either on previous Putnam exams or in the books Problemesol39uing Through Problems by LC Larson and Problemesol39uing Strategies by A Engel Problem 2 A farmer with 1000 feet of fencing wishes to fence a rectangular eld adjacent to a straight river Naturally7 no fence is needed along the river Use the arithmetic mean7geometricmean inequality to nd the dimensions of the eld that maximize its area Problem 3 Prove that for each positive integer n 171 1 n1 1 7 1 1 u ltlt am We proved this inequality last semester using the binomial theorem this time7 do it by showing that 1 11 is an increasing function Problem 4 Jensen7s inequality Let f R 7gt R be a twice differentiable function with f z 2 0 for all z 1f1 1 An are positive real numbers such that A1 An 17 then f S for any 11 1 1 1 In E R Deduce the general arithmetic mean7geometricmean inequality Problem 5 For which real numbers k does the inequality coshz S 62622 hold for all real 1 Problem 6 Show that if a b c are positive numbers with a b c 17 then 1 2 1 2 1 2 100 a 7 b 7 6 7 2 7 a b c 3 Problem 7 Show that if Ck lt Z gt for n gt 2 and 1 S k S n then ZJCk g n2 71 161 Problem 8 Show that for any integer n gt 17 1 1 lt 1 1 1 7 7 7 lt 1 7 7 lt 7 7 7 6 me n e 2ne Problem 9 Given n 2 2 which of the two numbers is larger a An exponential tower of n 27s or an exponential tower of n 7 1 37s b An exponential tower of n 37s or an exponential tower of n 7 1 47s Problem 10 Let Ly gt 07 and let 8 be the smallest of the numbers I y 117 1y Find the greatest possible value of s For which I y is this value assumed Problem 11 Suppose twenty disjoined squares lie inside a square of side 1 Prove that there are four squares among them such that the sum of the lengths of their sides does not exceed Problem 12 Find all positive integers n such that 3 4 n2 n3

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