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Convexity and concavity of a function

  1. Jan 23, 2008 #1
    here is the problem:
    we know that
    (1) f(x) is defined on [tex](-\infty, +\infty)[/tex], and f(x) has the second derivative everywhere,
    (2) [tex]lim_{|x|\rightarrow+\infty}(f(x)-|x|)=0[/tex],
    (3) there is [tex]x_0\in R[/tex] such that [tex]f(x_0)\leq0[/tex]
    how do we prove that f"(x) changes sign on [tex](-\infty, +\infty)[/tex]?

    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?
  2. jcsd
  3. Jan 25, 2008 #2
    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.
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