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Byju's Answer
Standard XII
Mathematics
Definition of Function
If f is a r...
Question
If
f
is a real-valued differentiable function such that
f
(
x
)
f
′
(
x
)
<
0
for all real
x
, then
A
f
(
x
)
must be a increasing function
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B
f
(
x
)
must be a decreasing function
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C
|
f
(
x
)
|
must be a increasing function
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D
|
f
(
x
)
|
must be a decreasing function
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Solution
The correct option is
D
|
f
(
x
)
|
must be a decreasing function
Given,
f
(
x
)
f
′
(
x
)
<
0
⇒
f
(
x
)
and
f
′
(
x
)
must be of opposite sign.
(i) Let
f
(
x
)
=
e
−
x
∴
f
′
(
x
)
−
e
−
x
⇒
f
(
x
)
>
0
and
f
′
(
x
)
<
0
,
∀
x
∈
R
(ii) Let
f
(
x
)
=
−
e
−
x
∴
f
′
(
x
)
=
e
−
x
⇒
f
(
x
)
<
0
and
f
′
(
x
)
>
0
,
∀
x
∈
R
But
|
f
(
x
)
|
=
|
±
e
−
x
|
=
e
−
x
in both cases
∴
d
d
x
|
f
(
x
)
|
−
e
−
x
<
0
in both case,
∀
⋅
x
∈
R
⇒
|
f
(
x
)
|
must be a decreasing function.
Suggest Corrections
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