1

I am dealing with the test of the OBM (Brasilian Math Olympiad), University level, 2016, phase 2.

As I've said at this topic (question 1), this other (question 2) and this (question 3), this (question 5) and this (question 4), I hope someone can help me to discuss this test. Thanks for any help.

The question 6 says:

Let $C, D>0$. We say a function $f:\mathbb{R}\rightarrow \mathbb{R}$ is beautiful if $f\in C^2$, $|x^3f(x)|\leq C$ and $|xf''(x)|\leq D$ for all $x$ with $|x|\geq1$.

(i) Prove that if $f$ is beautiful, given $\epsilon >0$, there's $x_0>0$ such that, for $|x|\geq x_0$, $|x^2f'(x)|<\sqrt{2CD}+\epsilon$.

(ii) Prove that, if $0<E<\sqrt{2CD}$, then exists a beautiful function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that, for all $x_0>0$, there's $x>x_0$ with $|x^2f'(x)|>E$.

I think there's something with Taylor's formula. I know also that $\lim_{x\to\pm\infty}\frac{f(x)}{x^2}=\lim_{x\to\pm\infty}f''(x)=0$... Right?

A important result is: "If $f$ is derivative in $\mathbb{R}$ and $f'$ is limited, so $f$ is lipschtiz."

Therefore, $f|_{\mathbb{R}-[-1,1]},f'|_{\mathbb{R}-[-1,1]}$ are lipschtiz.

More than an answer, I'd like to get some clues too. In general, I cannot know the way and results I should take. Should I take any way and try, try, try? What did you do when take a dificult question?

Thank you for a help and a blessed day.

Community
  • 1
Quiet_waters
  • 1,493
  • 1
    Yes, you can use some Taylor-type expansion (or integration). If you have seen the version without $x$ factors in front and over all $\mathbb{R}$, i,e, $\lVert u'\rVert\leq\sqrt{\lVert u\rVert\cdot\lVert u''\rVert}$ if $u\in C^2(\mathbb{R})$ bounded), try adapting that proof to this case. – user10354138 Nov 13 '18 at 02:28
  • Sorry because I didn't return, I was desconected of these questions for some days. Now, I'll try do what you've said. Thank you. – Quiet_waters Nov 28 '18 at 19:58

0 Answers0