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Rotwang
Posts:
1,678
From:
Swansea
Registered:
7/26/06


Re: Unbounded second derivative
Posted:
Dec 3, 2012 11:57 AM


On 03/12/2012 16:08, José Carlos Santos wrote: > On 03122012 11:17, José Carlos Santos wrote: > >> This is probably very simple, but I can't see it. :) Let f be a twice >> derivable function from [0,+oo[ into R such that: >> >> 1) lim_{x to +oo} f(x) = 0; >> >> 2) lim_{x to +oo} f'(x) does not exist. >> >> Prove that the function f'' is unbounded. > > Forget it! I've done it.
I was half way through a solution when you posted your reply, and it's a fun problem so I may as well finish it:
Since lim_{x > oo} f'(x) != 0, there exists d > 0 such that at least one of the sets {x  f'(x) > 2d} or {x  f'(x) < 2d} is unbounded; wolog let's suppose the first one is. Let M > 0, and let X be such that f(x) < d^2/2M whenever x > X. There exists x > X such that f'(x) > 2d. If f'(x) >= d for all y in [x, x + d/M] then we would have f(x + d/M) >= d^2/2M, so there must exist y such that x < y < x + d/M and f'(y) < d. By the mean value theorem there exists z in [x, y] such that
f''(z) = (f'(y)  f'(x))/(y  x) >= d/(d/M) = M.
Since this is true for any M, f'' is unbounded.
Did you find an easier way?
 I have made a thing that superficially resembles music:
http://soundcloud.com/eroneity/weberatedourowncrapiness



