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Mathematics
Problem
For
x
>
0
x>0
x
>
0
, let
F
(
x
)
=
x
2
k
(
x
)
F(x)=x^{2}k(x)
F
(
x
)
=
x
2
k
(
x
)
and
F
′
(
x
)
=
x
2
ln
x
+
4
x
F'(x)=x^{2}\ln x+4x
F
′
(
x
)
=
x
2
ln
x
+
4
x
. If
k
(
1
)
=
3
k(1)=3
k
(
1
)
=
3
, which of the following is
k
(
x
)
k(x)
k
(
x
)
? a)
x
3
ln
x
+
x
9
+
2
+
10
9
x
2
\dfrac{x}{3}\ln x+\dfrac{x}{9}+2+\dfrac{10}{9x^{2}}
3
x
ln
x
+
9
x
+
2
+
9
x
2
10
b)
ln
x
+
4
x
\ln x+\dfrac{4}{x}
ln
x
+
x
4
c)
x
3
ln
x
−
x
9
+
2
+
10
9
x
2
\dfrac{x}{3}\ln x-\dfrac{x}{9}+2+\dfrac{10}{9x^{2}}
3
x
ln
x
−
9
x
+
2
+
9
x
2
10
d)
x
3
ln
x
−
x
9
+
28
9
\dfrac{x}{3}\ln x-\dfrac{x}{9}+\dfrac{28}{9}
3
x
ln
x
−
9
x
+
9
28
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