(a) Use properties of quadratic functions to prove that 6(x - 1)² ≥² for all real x > 2. (b) Use mathematical induction and the inequality from part (a) to prove that 4-6 ≥ 5+1 + (n − 1)² for all integers n ≥ 2. (c) Let g(n) = 5+¹ + (n − 1)² and h(n) = 6". Using the inequality from part (b), prove that g(n) = O(h(n)). You need to give a rigorous proof derived directly from the definition of the O-notation, without using any theorems from class. (First, give a complete statement of the definition. Next, show how g(n) = O(h(n)) follows from this definition.)
(a) Use properties of quadratic functions to prove that 6(x - 1)² ≥² for all real x > 2. (b) Use mathematical induction and the inequality from part (a) to prove that 4-6 ≥ 5+1 + (n − 1)² for all integers n ≥ 2. (c) Let g(n) = 5+¹ + (n − 1)² and h(n) = 6". Using the inequality from part (b), prove that g(n) = O(h(n)). You need to give a rigorous proof derived directly from the definition of the O-notation, without using any theorems from class. (First, give a complete statement of the definition. Next, show how g(n) = O(h(n)) follows from this definition.)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 42E
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