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Mathematics
Mathematics
Problem
Let
F
:
R
→
R
F: \mathbb{R} \to \mathbb{R}
F
:
R
→
R
be a thrice differentiable function. Suppose that
F
(
1
)
=
0
F(1) = 0
F
(
1
)
=
0
,
F
(
3
)
=
−
4
F(3) = -4
F
(
3
)
=
−
4
, and
F
′
(
x
)
<
0
F'(x) < 0
F
′
(
x
)
<
0
for all
x
∈
(
1
,
3
)
x \in (1, 3)
x
∈
(
1
,
3
)
. Let
f
(
x
)
=
x
F
(
x
)
f(x) = xF(x)
f
(
x
)
=
x
F
(
x
)
for all
x
∈
R
x \in \mathbb{R}
x
∈
R
. The correct statement(s) is/are a.
f
′
(
1
)
<
0
f'(1) < 0
f
′
(
1
)
<
0
b.
f
(
2
)
<
0
f(2) < 0
f
(
2
)
<
0
c.
f
′
(
x
)
≠
0
f'(x) \neq 0
f
′
(
x
)
=
0
for any
x
∈
(
1
,
3
)
x \in (1, 3)
x
∈
(
1
,
3
)
d.
f
′
(
x
)
=
0
f'(x) = 0
f
′
(
x
)
=
0
for some
x
∈
(
1
,
3
)
x \in (1, 3)
x
∈
(
1
,
3
)
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