2. (This problem derives the general result that any polynomial function is of a lower asymptoticorder than any exponential function.)First, we prove a special case. Suppose that we have two functions, f(n) = n² and g(n) = 2".We will prove that f(n) = O(g(n)).• port (a).(c) Show that, for any n ≥ no, it must be the case that f(n) ≤ cog(n) = f(n+1) ≤cog(n+1). Precisely state the argument by induction that n² = 0(2¹).Now, define f(n) = nk where k ≥ 1, g(n) = a", where a ≥ 1.(d) (Slightly harder). Argue that, for any k and a, there exists some n₁ such that: (1₁+¹)* <a.

To show that f(n) = n^2 = O(g(n)) = 2^n, we need to find a constant c and a value n0 such that for…

2. (This problem derives the general result that any polynomial function is of a lower asymptotic order than any exponential function.) First, we prove a special case. Suppose that we have two functions, f(n) = n² and g(n) = 2". We will prove that f(n) = O(g(n)). por (a). (c) Show that, for any n2 no, it must be the case that f(n) ≤ cog(n) = f(n+1) ≤ cog(n+1). Precisely state the argument by induction that n² = O(2"). Now, define f(n) = nk where k≥ 1, g(n) = a", where a ≥ 1.

2. (This problem derives the general result that any polynomial function is of a lower asymptotic order than any exponential function.) First, we prove a special case. Suppose that we have two functions, f(n) = n² and g(n) = 2". We will prove that f(n) = O(g(n)). por (a). (c) Show that, for any n2 no, it must be the case that f(n) ≤ cog(n) = f(n+1) ≤ cog(n+1). Precisely state the argument by induction that n² = O(2"). Now, define f(n) = nk where k≥ 1, g(n) = a", where a ≥ 1.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter2: Mathematics For Microeconomics

Section: Chapter Questions

Problem 2.13P

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