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Undergraduate contests
Brazil Undergrad MO
2016 Brazil Undergrad MO
6
6
Part of
2016 Brazil Undergrad MO
Problems
(1)
Inequalities and pretty functions
Source: 38th Brazilian Undergrad MO (2016) - Second Day, Problem 6
11/25/2016
Let it
C
,
D
>
0
C,D > 0
C
,
D
>
0
. We call a function
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
pretty if
f
f
f
is a
C
2
C^2
C
2
-class,
∣
x
3
f
(
x
)
∣
≤
C
|x^3f(x)| \leq C
∣
x
3
f
(
x
)
∣
≤
C
and
∣
x
f
′
′
(
x
)
∣
≤
D
|xf''(x)| \leq D
∣
x
f
′′
(
x
)
∣
≤
D
.[list='i'] [*] Show that if
f
f
f
is pretty, then, given
ϵ
≥
0
\epsilon \geq 0
ϵ
≥
0
, there is a
x
0
≥
0
x_0 \geq 0
x
0
≥
0
such that for every
x
x
x
with
∣
x
∣
≥
x
0
|x| \geq x_0
∣
x
∣
≥
x
0
, we have
∣
x
2
f
′
(
x
)
∣
<
2
C
D
+
ϵ
|x^2f'(x)| < \sqrt{2CD}+\epsilon
∣
x
2
f
′
(
x
)
∣
<
2
C
D
+
ϵ
. [*] Show that if
0
<
E
<
2
C
D
0 < E < \sqrt{2CD}
0
<
E
<
2
C
D
then there is a pretty function
f
f
f
such that for every
x
0
≥
0
x_0 \geq 0
x
0
≥
0
there is a
x
>
x
0
x > x_0
x
>
x
0
such that
∣
x
2
f
′
(
x
)
∣
>
E
|x^2f'(x)| > E
∣
x
2
f
′
(
x
)
∣
>
E
.
real analysis