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Inequalities and pretty functions

Source: 38th Brazilian Undergrad MO (2016) - Second Day, Problem 6

November 25, 2016
real analysis

Problem Statement

Let it C,D>0C,D > 0. We call a function f:R→Rf:\mathbb{R} \rightarrow \mathbb{R} pretty if ff is a C2C^2-class, ∣x3f(x)∣≤C|x^3f(x)| \leq C and ∣xf′′(x)∣≤D|xf''(x)| \leq D.
[list='i'] [*] Show that if ff is pretty, then, given ϵ≥0\epsilon \geq 0, there is a x0≥0x_0 \geq 0 such that for every xx with ∣x∣≥x0|x| \geq x_0, we have ∣x2f′(x)∣<2CD+ϵ|x^2f'(x)| < \sqrt{2CD}+\epsilon. [*] Show that if 0<E<2CD0 < E < \sqrt{2CD} then there is a pretty function ff such that for every x0≥0x_0 \geq 0 there is a x>x0x > x_0 such that ∣x2f′(x)∣>E|x^2f'(x)| > E.
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