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Putnam
1946 Putnam
A1
Putnam 1946 A1
Putnam 1946 A1
Source: Putnam 1946
March 10, 2022
Putnam
function
Problem Statement
Suppose that the function
f
(
x
)
=
a
x
2
+
b
x
+
c
f(x)=a x^2 +bx+c
f
(
x
)
=
a
x
2
+
b
x
+
c
, where
a
,
b
,
c
a,b,c
a
,
b
,
c
are real, satisfies the condition
∣
f
(
x
)
∣
≤
1
|f(x)|\leq 1
∣
f
(
x
)
∣
≤
1
for
∣
x
∣
≤
1
|x|\leq1
∣
x
∣
≤
1
. Prove that
∣
f
′
(
x
)
∣
≤
4
|f'(x)|\leq 4
∣
f
′
(
x
)
∣
≤
4
for
∣
x
∣
≤
1
|x|\leq1
∣
x
∣
≤
1
.
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