(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.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:(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)=5n+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.)
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