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Byju's Answer
Standard XIII
Mathematics
Monotonically Decreasing Functions
Let fx be a n...
Question
Let
f
(
x
)
be a non-negative function. If
f
′
(
x
)
cos
x
≤
f
(
x
)
sin
x
,
∀
x
≥
0
,
then value of
f
(
5
π
3
)
is
A
1
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B
1
2
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C
0
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D
√
3
2
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Solution
The correct option is
C
0
f
′
(
x
)
cos
x
≤
f
(
x
)
sin
x
⇒
f
′
(
x
)
cos
x
−
f
(
x
)
sin
x
≤
0
⇒
d
d
x
(
f
(
x
)
cos
x
)
≤
0
∴
g
(
x
)
=
f
(
x
)
cos
x
is decreasing
∀
x
≥
0
⇒
g
(
x
)
≤
g
(
0
)
⇒
f
(
x
)
cos
x
≤
f
(
0
)
cos
0
⇒
f
(
x
)
cos
x
≤
f
(
0
)
We know that
5
π
3
≥
π
2
⇒
g
(
5
π
3
)
≤
g
(
π
2
)
(
∵
g
(
x
)
is decreasing
)
⇒
g
(
5
π
3
)
≤
0
[
∵
g
(
π
2
)
=
0
]
⇒
f
(
5
π
3
)
×
cos
(
5
π
3
)
≤
0
⇒
f
(
5
π
3
)
×
cos
(
π
3
)
≤
0
⇒
f
(
5
π
3
)
≤
0
but
f
(
x
)
≥
0
(Given in the question)
So,
f
(
5
π
3
)
=
0
Suggest Corrections
0
Similar questions
Q.
Let
f
(
x
)
be a non-negative function. If
f
′
(
x
)
cos
x
≤
f
(
x
)
sin
x
,
∀
x
≥
0
,
then value of
f
(
5
π
3
)
is
Q.
Let g (x),
x
≥
0
, be a non-negative continuous function and let
F
(
x
)
=
∫
x
0
f
(
t
)
d
t
,
x
≥
0
. If for some c > 0, f(x)
≤
c F(x) for all
x
≥
0
, then
Q.
Let
f
(
x
)
be a non-negative function. If
f
′
(
x
)
cos
x
≤
f
(
x
)
sin
x
,
∀
x
≥
0
,
then value of
f
(
5
π
3
)
is
Q.
Let
f
(
x
)
be a non-negative differentiable function on
[
0
,
∞
)
such that
f
(
0
)
=
0
and
f
′
(
x
)
≤
2
f
(
x
)
for all
x
>
0
. Then, on
[
0
,
∞
)
.
Q.
Let
F
(
x
)
be a non-negative continuous function and
F
(
x
)
=
x
∫
0
f
(
x
)
d
x
∀
x
≥
0.
If for some
c
>
0
,
f
(
x
)
≤
c
F
(
x
)
for all
x
≥
0
,
then which of the following is/are always correct ?
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