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Sweden Contests
Swedish Mathematical Competition
1976 Swedish Mathematical Competition
5
5
Part of
1976 Swedish Mathematical Competition
Problems
(1)
f(1) <=4/3 if (1+f(x))f''(x)=1+x , f(0)=1 , f'(0)=0
Source: 1976 Swedish Mathematical Competition p5
3/26/2021
f
(
x
)
f(x)
f
(
x
)
is defined for
x
≥
0
x \geq 0
x
≥
0
and has a continuous derivative. It satisfies
f
(
0
)
=
1
f(0)=1
f
(
0
)
=
1
,
f
′
(
0
)
=
0
f'(0)=0
f
′
(
0
)
=
0
and
(
1
+
f
(
x
)
)
f
′
′
(
x
)
=
1
+
x
(1+f(x))f''(x)=1+x
(
1
+
f
(
x
))
f
′′
(
x
)
=
1
+
x
. Show that
f
f
f
is increasing and that
f
(
1
)
≤
4
/
3
f(1) \leq 4/3
f
(
1
)
≤
4/3
.
analysis
inequalities
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