Convexity and concavity of a function

[y_lindsay]

here is the problem:
we know that
(1) f(x) is defined on (-\infty, +\infty), and f(x) has the second derivative everywhere,
(2) lim_{|x|\rightarrow+\infty}(f(x)-|x|)=0,
(3) there is x_0\in R such that f(x_0)\leq0
how do we prove that f"(x) changes sign on (-\infty, +\infty)?

I can imagine that the graph of f(x) can't be always convex or concave on the real line, but how do I put it in hard argument?

 

[birulami]

Funny you did not get an answer yet, as I consider this an interesting problem. The conclusion seems to be so obvious when you try to draw the graph.