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Mathematics
Problem
A function
f
(
x
)
f(x)
f
(
x
)
satisfies the relation
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
x
y
(
x
+
y
)
,
∀
x
,
y
∈
R
f(x+y) = f(x) + f(y) + xy(x+y), \forall x, y \in \mathbb{R}
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
x
y
(
x
+
y
)
,
∀
x
,
y
∈
R
. If
f
′
(
0
)
=
−
1
f'(0) = -1
f
′
(
0
)
=
−
1
, then a.
f
(
x
)
f(x)
f
(
x
)
is a polynomial function b.
f
(
x
)
f(x)
f
(
x
)
is an exponential function c.
f
(
x
)
f(x)
f
(
x
)
is twice differentiable for all
x
∈
R
x \in \mathbb{R}
x
∈
R
d.
f
′
(
3
)
=
8
f'(3) = 8
f
′
(
3
)
=
8
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