如何找到功能的上限

Question

How to find the upper bound (Big O) for f(n) = 2n^3-2n^2. Want to know the C and n.

I tried below.

1. f(n)<=C*g(n)

2. 2n^3-2n^2 <= C*n^3

   My understanding of finding "C" is, consider only higher polynomial degree. So..

3. 2n^3 <= C*n^3 ---> (omitted 2n^2)

4. Thus arrived the value of C as 3.

But the answer given in my book is C = 2.

question 1: Please clarify how C will be 2.

question 2: how to find the value of n.

Answer 1

The inequality you have to prove is:

2n^3-2n^2 <= C*n^3 for all n >= n0.

Actually, any C >= 2 works. n0 in this case can be taken to anything >= 0.