- #1
y_lindsay
- 17
- 0
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?
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?