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Byju's Answer
Standard XII
Mathematics
Strictly Increasing Functions
Which of the ...
Question
Which of the following functions is always increasing?
A
x
+
sin
2
x
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B
x
−
sin
2
x
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C
2
x
+
sin
3
x
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D
2
x
−
sin
x
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Solution
The correct option is
D
2
x
−
sin
x
A function is said to be always increasing if its derivative is always greater than zero. i.e,
d
y
d
x
>
0
(
a
)
y
=
x
+
s
i
n
2
x
d
y
d
x
=
1
+
2
c
o
s
2
x
⇒
−
1
≤
(
1
+
2
c
o
s
2
x
)
≤
3
∵
(
−
1
≤
c
o
s
2
x
≤
1
)
(
b
)
y
=
x
−
s
i
n
2
x
d
y
d
x
=
1
−
2
c
o
s
2
x
⇒
−
1
≤
(
1
−
2
c
o
s
2
x
)
≤
3
∵
(
−
1
≤
c
o
s
2
x
≤
1
)
(
c
)
y
=
2
x
+
s
i
n
3
x
d
y
d
x
=
3
+
3
c
o
s
3
x
⇒
0
≤
(
3
+
3
c
o
s
3
x
)
≤
6
∵
(
−
1
≤
c
o
s
3
x
≤
1
)
(
d
)
y
=
2
x
−
s
i
n
x
d
y
d
x
=
2
−
c
o
s
x
⇒
1
≤
(
2
−
c
o
s
x
)
≤
3
∵
(
−
1
≤
c
o
s
x
≤
1
)
Clearly in the case of (d) dy/dx > 0
∴
2
x
−
sin
x
is always increasing.
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