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Byju's Answer
Standard XII
Mathematics
Definition of Function
Prove that ...
Question
Prove that
y
=
4
sin
θ
2
+
cos
θ
−
θ
is an increasing function in
[
0
,
π
2
]
.
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Solution
Let
f
(
θ
)
=
4
sin
θ
2
+
cos
θ
−
θ
,
θ
∈
[
0
,
π
2
]
Now
f
′
(
θ
)
=
4
(
1
+
2
cos
θ
)
(
2
+
cos
θ
)
2
−
1
The function is continuous in
[
0
,
π
2
]
and differentiable in
(
0
,
π
2
)
.
Then from Lagrange's Mean Value Theorem we get, there exist
c
∈
(
0
,
π
2
)
such that,
f
(
π
2
)
−
f
(
0
)
π
2
=
f
′
(
c
)
>
0
So we have
f
′
(
θ
)
>
0
,
∀
θ
∈
(
0
,
π
2
)
.
Now
f
′
(
0
)
=
1
3
and
f
′
(
π
2
)
=
0
.
So,
f
′
(
θ
)
≥
0
∀
θ
∈
[
0
,
π
2
]
.
This gives the function
f
(
θ
)
is monotone increasing in
[
0
,
π
2
]
.
So
y
is an increasing function in that interval.
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Definition of Function
Standard XII Mathematics
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