264
You visited us
264
times! Enjoying our articles?
Unlock Full Access!
Byju's Answer
Standard XII
Mathematics
Rolle's Theorem
In which of t...
Question
In which of the following functions is Rolle's theorem applicable
A
f
(
x
)
=
{
x
,
0
≤
x
<
1
0
,
x
=
1
on
[
0
,
1
]
No worries! We‘ve got your back. Try BYJU‘S free classes today!
B
f
(
x
)
=
⎧
⎨
⎩
sin
x
x
,
−
π
≤
x
<
0
0
,
x
=
0
on
[
−
π
,
0
]
No worries! We‘ve got your back. Try BYJU‘S free classes today!
C
f
(
x
)
=
x
2
−
x
−
6
x
−
1
on
[
−
2
,
3
]
No worries! We‘ve got your back. Try BYJU‘S free classes today!
D
f
(
x
)
=
⎧
⎨
⎩
x
3
−
2
x
2
−
5
x
+
6
x
−
1
,
i
f
x
≠
1
−
6
,
i
f
x
=
1
on
[
−
2
,
3
]
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
Open in App
Solution
The correct option is
D
f
(
x
)
=
⎧
⎨
⎩
x
3
−
2
x
2
−
5
x
+
6
x
−
1
,
i
f
x
≠
1
−
6
,
i
f
x
=
1
on
[
−
2
,
3
]
from lagranges theorem
only option D satisfies all condition.
Suggest Corrections
0
Similar questions
Q.
Consider the function for
x
=
[
−
2
,
3
]
,
f
(
x
)
=
⎧
⎪
⎨
⎪
⎩
x
3
−
2
x
2
−
5
x
+
6
x
−
1
,
i
f
x
≠
1
−
6
i
f
x
=
1
then
Q.
Consider the function for
x
∈
[
−
2
,
3
]
f
(
x
)
=
⎧
⎨
⎩
−
6
;
x
=
1
x
3
−
2
x
2
−
5
x
+
6
x
−
1
;
x
≠
1
. The value of c obtained by applying Rolle's theorem for which
f
′
(
c
)
=
0
is
Q.
Verify Lagrange's mean value theorem for the following functions on the indicated intervals. In each case find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem
(i) f(x) = x
2
− 1 on [2, 3]
(ii) f(x) = x
3
− 2x
2
− x + 3 on [0, 1]
(iii) f(x) = x(x −1) on [1, 2]
(iv) f(x) = x
2
− 3x + 2 on [−1, 2]
(v) f(x) = 2x
2
− 3x + 1 on [1, 3]
(vi) f(x) = x
2
− 2x + 4 on [1, 5]
(vii) f(x) = 2x − x
2
on [0, 1]
(viii) f(x) = (x − 1)(x − 2)(x − 3) on [0, 4]
(ix)
f
x
=
25
-
x
2
on [−3, 4]
(x) f(x) = tan
−
1
x on [0, 1]
(xi)
f
x
=
x
+
1
x
on
[
1
,
3
]
(xii) f(x) = x(x + 4)
2
on [0, 4]
(xiii)
f
x
=
x
2
-
4
on
[
2
,
4
]
(xiv) f(x) = x
2
+ x − 1 on [0, 4]
(xv) f(x) = sin x − sin 2x − x on [0, π]
(xvi) f(x) = x
3
− 5x
2
− 3x on [1, 3]
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]