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Byju's Answer
Standard XII
Mathematics
Definition of Functions
The value of ...
Question
The value of
f
(
x
)
=
(
π
4
sin
2
x
)
,
x
∈
R
lies in the interval
A
[
0
,
1
]
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B
[
0
,
π
4
]
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C
[
0
,
π
2
]
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D
[
π
4
,
1
]
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Solution
The correct option is
C
[
0
,
π
4
]
sin
2
x
lies between
0
to
1
for all values of
x
∴
f
(
x
)
lies between
[
0
,
π
4
]
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0
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Q.
The value of
x
in the interval
[
0
,
2
π
]
for which
4
sin
2
x
−
8
sin
x
+
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≤
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is
Q.
If
θ
∈
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,
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)
, then the value of
cos
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Q.
If the function
f
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⎧
⎪ ⎪
⎨
⎪ ⎪
⎩
x
+
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2
s
i
n
x
,
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c
o
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,
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≤
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b
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i
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Q.
Verify Rolle's theorem for each of the following functions on the indicated intervals
(i) f(x) = cos 2 (x − π/4) on [0, π/2]
(ii) f(x) = sin 2x on [0, π/2]
(iii) f(x) = cos 2x on [−π/4, π/4]
(iv) f(x) = e
x
sin x on [0, π]
(v) f(x) = e
x
cos x on [−π/2, π/2]
(vi) f(x) = cos 2x on [0, π]
(vii) f(x) =
sin
x
e
x
on 0 ≤ x ≤ π
(viii) f(x) = sin 3x on [0, π]
(ix) f(x) =
e
1
-
x
2
on [−1, 1]
(x) f(x) = log (x
2
+ 2) − log 3 on [−1, 1]
(xi) f(x) = sin x + cos x on [0, π/2]
(xii) f(x) = 2 sin x + sin 2x on [0, π]
(xiii)
f
x
=
x
2
-
sin
π
x
6
on
[
-
1
,
0
]
(xiv)
f
x
=
6
x
π
-
4
sin
2
x
on
[
0
,
π
/
6
]
(xv) f(x) = 4
sin
x
on [0, π]
(xvi) f(x) = x
2
− 5x + 4 on [1, 4]
(xvii) f(x) = sin
4
x + cos
4
x on
0
,
π
2
(xviii) f(x) = sin x − sin 2x on [0, π]