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Putnam
2019 Putnam
A6
A6
Part of
2019 Putnam
Problems
(1)
Putnam 2019 A6
Source:
12/10/2019
Let
g
g
g
be a real-valued function that is continuous on the closed interval
[
0
,
1
]
[0,1]
[
0
,
1
]
and twice differentiable on the open interval
(
0
,
1
)
(0,1)
(
0
,
1
)
. Suppose that for some real number
r
>
1
r>1
r
>
1
,
lim
x
→
0
+
g
(
x
)
x
r
=
0.
\lim_{x\to 0^+}\frac{g(x)}{x^r} = 0.
x
→
0
+
lim
x
r
g
(
x
)
=
0.
Prove that either
lim
x
→
0
+
g
′
(
x
)
=
0
or
lim sup
x
→
0
+
x
r
∣
g
′
′
(
x
)
∣
=
∞
.
\lim_{x\to 0^+}g'(x) = 0\qquad\text{or}\qquad \limsup_{x\to 0^+}x^r|g''(x)|= \infty.
x
→
0
+
lim
g
′
(
x
)
=
0
or
x
→
0
+
lim
sup
x
r
∣
g
′′
(
x
)
∣
=
∞.
Putnam
Putnam 2019