problemset3

X1 x x13 49 let 0 y x show that xy xy 2 log x

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Unformatted text preview: lity holding if and only if a = 2 and x = 1. �n 1 41. Show that if xi ∈ 0 for i = 1, 2 . . . , n and i=1 1+xi � 1, then �n −xi � 1. i=1 2 � 42. Let 0 � ai < 1 for i = 1, 2, . . . , n, and put n=1 ai = A. Show that i n � i=1 ai nA ∈ , 1 − ai n−A with equality if and only if all the ai are equal. 43. Show that for n ∈ 2, n � � n� i=0 i � � 2n − 2 n−1 �n−1 . 44. Let b1 , . . . , bn be any rearrangement of the positive numbers a1 , . . . , an . Show that a1 an +···+ ∈ n. b1 bn 10 45. Given that �n i=1 bi = b with each bi a nonnegative number, show that n−1 � j =1 bj bj +1 � b2 . 4 46. Let n ∈ 2 and 0 < x1 < x2 < · · · < xn � 1. Show that n n � xk k =1 n � k =1 xk + nx1 x2 · · · xn ∈ n � k =1 1 . 1 + xk 47. Let f b e a continuous function on the interval [0, 1] such that 0 < m � f (x) � M for all x in [0, 1]. Show that �� 1 � �� 1 � dx (m + M )2 f (x)dx � . 4mM 0 f (x) 0 48. Let x > 0 and x = 1. Show that ≤ log x 1 �� x−1 x log x 1 + x1/3 � . x−1 x + x1/3 49. Let 0 < y < x. Show that x+y x−y > . 2 log x − log y 50. Let x > 0. Show that � � 2 1 1 < log x + <� . 2x + 1 x x2 + x 1 1 51. Let Sn = 1 + 2 + 1 + · · · + n . Show that 3 � � � 1/n n (1 + n) − 1 < Sn < n 1 − (n + 1)−1/n − 11 1 n+1 � . 52. Let x > 0 and y > 0. Show that 1 − e−x−y 1 1 − �. −x )(1 − e−y ) (x + y )(1 − e xy 12 53. Let a, b, c, d, e, and f be nonegative real numbers satisfying a + b � e and c + d � f . Show that � ac + � 54. Show that for x > 0 and x = 1, ≤ 0� bd � � ef . x log x 1 �. 2−1 2 x 55. Show that for x > 0, x(2 + cos x) > 3 sin x. 56. Show that for 0 < x < � /2, 2 sin x + tan x > 3x. 57. Let x > 0, x = 1, and suppose that n is a positive integer. Show that ≤ x+ 1 x−1 > 2n n . n x x −1...
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This note was uploaded on 01/22/2014 for the course MATH 18.S34 taught by Professor Hartleyrogers during the Fall '07 term at MIT.

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