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If a function satisfies
MATHEMATICS · Single correct · AoPS · Mathematics · Functions · Functions
Problem
If a function satisfies
(
x
−
y
)
f
(
x
+
y
)
−
(
x
+
y
)
f
(
x
−
y
)
=
2
(
x
2
y
−
y
3
)
(x-y)f(x+y) - (x+y)f(x-y) = 2(x^2y - y^3)
(
x
−
y
)
f
(
x
+
y
)
−
(
x
+
y
)
f
(
x
−
y
)
=
2
(
x
2
y
−
y
3
)
,
∀
x
,
y
∈
R
\forall x, y \in R
∀
x
,
y
∈
R
and
f
(
1
)
=
2
f(1) = 2
f
(
1
)
=
2
, then (1)
f
(
x
)
f(x)
f
(
x
)
must be polynomial function (2)
f
(
3
)
=
12
f(3) = 12
f
(
3
)
=
12
(3)
f
(
0
)
=
0
f(0) = 0
f
(
0
)
=
0
(4)
f
(
x
)
f(x)
f
(
x
)
may not be differentiable
Related theory
Functions
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