Hello everyone, this is a question about the question number 22. It is in response to the previous here:

http://www.mathematicsgre.com/viewtopic.php?f=1&t=157&hilit=22

gaucho85 says he represents the functions f''(x), f'(x), and f(x) as x^3, x^2, and x respectively. But i don't see how he can do that. If f''(x) is x^3, then f'(x) should be (x^4)/4, and f(x) should be (x^5)/20. Is that right? And if i am wrong i am asking for a little help about why you can assign the f(x) function and its derivatives like that.

Then in the response there is this:

III is the answer since you pick up two functions

h1, h2 satisfy

h1''=h1+1

h2''=h2+1

then (h1+h2)''=h1''+h2''= h1+1+h2+1 =(h1+h2)+2 != (h1+h2)+1

And i think that just because you pick two functions h1 and h2 that both satisfy the equation for choice 3, it doesn't mean that (h1 + h2)'' should equal (h1+h2) +1. Is this right?

So basically i am asking if these two ways of solving the problem are actually correct, because i am a little skeptical. Also i am a little rusty and am trying to study for the GRE, so i would love a little help. Thanks a lot!

Also, there is another post already about this problem and it seems that it contains the best way to solve this problem. You just have to use the fact that for the sets to be subspaces of the real numbers then they must contain the 0 vector and answer 3 does not.