Picky questions about Big-O problem (Solution included).

[s3a]

Homework Statement

The problem and its solution is attached.

Homework Equations

Big-O notation.

The Attempt at a Solution

I get how to do this problem. The reason I'm making this post is for little details about the constants c and n_0. To my knowledge both c and n_0 have to be greater than 0. For n_0, if you use the definition of natural numbers which excludes 0 then I have no problem. On the other hand, the constant c being any real implies that it can be 0 or a negative number as well which doesn't fit in well with what I have learned.

Am I wrong or is the solution wrong? What are the subtleties here?

 

[NascentOxygen]

O() is used for comparing the computations involved in computer algorithms, where c and n_o both need to be >0 for it to make sense. Beyond this, I can't say any more.

Seems to be a typo, that cn^1.5 should be cn^2.5 surely?

 

[s3a]

Yes, that n^(1.5) is a typo but it's not what I'm concerned about. ;)

Ok so, it seems that I must generally either make c > 0 or just let it be any constant including zero (it would technically work for a function that equals 0 and is big O of some other function where c = 0) or take the absolute value of the first function as well as the constant times the second function (which could be any real).

This problem puts no absolute value nor does it restrict c > 0 so having said that, the problem is incomplete, right?