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1. (10) An m x n rectangular array of distinct real numbers has the property that the numbers in each row increase from left to right. The entries in each column, individually, are rearranged so that the numbers in each column increase from top to bottom. Prove that in the final array, the numbers in each row will increase from left to right. 2. (11) Determine distinct positive integers a, b, c, d, e such that the five numbers a, b², c³, d4, e5 constitute an arithmetic progression. (The difference between adjacent pairs is the same.) 3. (10) Prove that the set {1,2,..., n} can be partitioned into k sub- sets with the same sum if and only if k divides 1½n(n + 1) and n≥ 2k - 1. 4. (6) Suppose that f(x) is a continuous real-valued function defined on the interval [0, 1] that is twice differentiable on (0, 1) and satisfies (i) f(0) = 0 and (ii) f"(x) > 0 for 0 < x < 1. (a) Prove that there exists a number a for which 0 < a < 1 and f'(a) < f(1); (b) Prove that there exists a unique number b for which a

1. (10) An m x n rectangular array of distinct real numbers has the property that the numbers in each row increase from left to right. The entries in each column, individually, are rearranged so that the numbers in each column increase from top to bottom. Prove that in the final array, the numbers in each row will increase from left to right. 2. (11) Determine distinct positive integers a, b, c, d, e such that the five numbers a, b², c³, d4, e5 constitute an arithmetic progression. (The difference between adjacent pairs is the same.) 3. (10) Prove that the set {1,2,..., n} can be partitioned into k sub- sets with the same sum if and only if k divides 1½n(n + 1) and n≥ 2k - 1. 4. (6) Suppose that f(x) is a continuous real-valued function defined on the interval [0, 1] that is twice differentiable on (0, 1) and satisfies (i) f(0) = 0 and (ii) f"(x) > 0 for 0 < x < 1. (a) Prove that there exists a number a for which 0 < a < 1 and f'(a) < f(1); (b) Prove that there exists a unique number b for which a

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 92E
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1. (10) An m x n rectangular array of distinct real numbers has the property that the numbers in each row increase from left to right. The entries in each column, individually, are rearranged so that the numbers in each column increase from top to bottom. Prove that in the final array, the numbers in each row will increase from left to right. 2. (11) Determine distinct positive integers a, b, c, d, e such that the five numbers a, b², c³, d4, e5 constitute an arithmetic progression. (The difference between adjacent pairs is the same.) 3. (10) Prove that the set {1,2,..., n} can be partitioned into k sub- sets with the same sum if and only if k divides 1½n(n + 1) and n≥ 2k - 1. 4. (6) Suppose that f(x) is a continuous real-valued function defined on the interval [0, 1] that is twice differentiable on (0, 1) and satisfies (i) f(0) = 0 and (ii) f"(x) > 0 for 0 < x < 1. (a) Prove that there exists a number a for which 0 < a < 1 and f'(a) < f(1); (b) Prove that there exists a unique number b for which a <b<1 and f'(a) = f(b)/b. = 5. (6) For x 1 and x 0, let - f(x)= −8[1 − (1 − x) 1/2]3 - = x2 (a) Prove that limo f(x) exists. Take this as the value of f(0). (b) Determine the smallest closed interval that contains all values assumed by f(x) on its domain. (c) Prove that f(f(f(x))) = f(x) for all x ≤ 1. 6. (4) Let h(n) denote the number of finite sequences {a1, a2,. of positive integers exceeding 1 for which k ≥ 1, a₁ ≥ a2 ≥ … … · ≥ ak ak} >
Transcribed Image Text:1. (10) An m x n rectangular array of distinct real numbers has the property that the numbers in each row increase from left to right. The entries in each column, individually, are rearranged so that the numbers in each column increase from top to bottom. Prove that in the final array, the numbers in each row will increase from left to right. 2. (11) Determine distinct positive integers a, b, c, d, e such that the five numbers a, b², c³, d4, e5 constitute an arithmetic progression. (The difference between adjacent pairs is the same.) 3. (10) Prove that the set {1,2,..., n} can be partitioned into k sub- sets with the same sum if and only if k divides 1½n(n + 1) and n≥ 2k - 1. 4. (6) Suppose that f(x) is a continuous real-valued function defined on the interval [0, 1] that is twice differentiable on (0, 1) and satisfies (i) f(0) = 0 and (ii) f"(x) > 0 for 0 < x < 1. (a) Prove that there exists a number a for which 0 < a < 1 and f'(a) < f(1); (b) Prove that there exists a unique number b for which a <b<1 and f'(a) = f(b)/b. = 5. (6) For x 1 and x 0, let - f(x)= −8[1 − (1 − x) 1/2]3 - = x2 (a) Prove that limo f(x) exists. Take this as the value of f(0). (b) Determine the smallest closed interval that contains all values assumed by f(x) on its domain. (c) Prove that f(f(f(x))) = f(x) for all x ≤ 1. 6. (4) Let h(n) denote the number of finite sequences {a1, a2,. of positive integers exceeding 1 for which k ≥ 1, a₁ ≥ a2 ≥ … … · ≥ ak ak} >
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