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Putnam
2018 Putnam
A5
A5
Part of
2018 Putnam
Problems
(1)
Putnam 2018 A5
Source:
12/2/2018
Let
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
be an infinitely differentiable function satisfying
f
(
0
)
=
0
f(0) = 0
f
(
0
)
=
0
,
f
(
1
)
=
1
f(1) = 1
f
(
1
)
=
1
, and
f
(
x
)
≥
0
f(x) \ge 0
f
(
x
)
≥
0
for all
x
∈
R
x \in \mathbb{R}
x
∈
R
. Show that there exist a positive integer
n
n
n
and a real number
x
x
x
such that
f
(
n
)
(
x
)
<
0
f^{(n)}(x) < 0
f
(
n
)
(
x
)
<
0
.
Putnam
Putnam 2018
Taylor series
college contests
real analysis