2. Suppose that the functions f1, f2, 91, 92 : N → R20 are such that f₁ = O(91) and ƒ2 € О(92). Prove that (fi + ƒ₂) € ☹(max{91, 92}). Here (f1f2)(n) = fi(n) + f2(n) and max{91, 92}(n) = max{91(n), 92(n)}. 3. Let nЄ N\{0}. Describe the largest set of values n for which you think 2n < n!. Use induction to prove that your description is correct. Here m! stands for m factorial, the product of first m positive integers. 4. Prove that log2 n! = O(n log n).

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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2. Suppose that the functions f1, f2, 91, 92 : N → R20 are such that f₁ = O(91) and ƒ2 € О(92).
Prove that (fi + ƒ₂) € ☹(max{91, 92}).
Here (f1f2)(n) = fi(n) + f2(n) and max{91, 92}(n) = max{91(n), 92(n)}.
3. Let nЄ N\{0}. Describe the largest set of values n for which you think 2n < n!. Use induction to
prove that your description is correct.
Here m! stands for m factorial, the product of first m positive integers.
4. Prove that log2 n! = O(n log n).
Transcribed Image Text:2. Suppose that the functions f1, f2, 91, 92 : N → R20 are such that f₁ = O(91) and ƒ2 € О(92). Prove that (fi + ƒ₂) € ☹(max{91, 92}). Here (f1f2)(n) = fi(n) + f2(n) and max{91, 92}(n) = max{91(n), 92(n)}. 3. Let nЄ N\{0}. Describe the largest set of values n for which you think 2n < n!. Use induction to prove that your description is correct. Here m! stands for m factorial, the product of first m positive integers. 4. Prove that log2 n! = O(n log n).
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