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IMC
2009 IMC
2
IMC 2009 Day 2 P2
IMC 2009 Day 2 P2
Source:
July 16, 2014
function
inequalities
IMC
college contests
Integral inequality
Problem Statement
Suppose
f
:
R
→
R
f:\mathbb{R}\to \mathbb{R}
f
:
R
→
R
is a two times differentiable function satisfying
f
(
0
)
=
1
,
f
′
(
0
)
=
0
f(0)=1,f^{\prime}(0)=0
f
(
0
)
=
1
,
f
′
(
0
)
=
0
and for all
x
∈
[
0
,
∞
)
x\in [0,\infty)
x
∈
[
0
,
∞
)
, it satisfies
f
′
′
(
x
)
−
5
f
′
(
x
)
+
6
f
(
x
)
≥
0
f^{\prime \prime}(x)-5f^{\prime}(x)+6f(x)\ge 0
f
′′
(
x
)
−
5
f
′
(
x
)
+
6
f
(
x
)
≥
0
Prove that, for all
x
∈
[
0
,
∞
)
x\in [0,\infty)
x
∈
[
0
,
∞
)
,
f
(
x
)
≥
3
e
2
x
−
2
e
3
x
f(x)\ge 3e^{2x}-2e^{3x}
f
(
x
)
≥
3
e
2
x
−
2
e
3
x
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