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Byju's Answer
Standard XII
Mathematics
Factorization Method Form to Remove Indeterminate Form
Is Rolle's th...
Question
Is Rolle's theorem applicable for the function
f
(
x
)
=
x
3
−
6
x
2
+
11
x
−
6
?
Open in App
Solution
Here, we have
f
(
x
)
=
x
3
−
6
x
2
+
11
x
−
6
,
i
f
f
(
x
)
=
0
Then
x
3
−
6
x
2
+
11
x
−
6
x
=
0
⇒
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
=
0
∴
x
=
1
,
2
,
3.
=
f
(
1
)
=
0
=
f
(
2
)
=
f
(
3
)
Also
f
′
(
x
)
=
3
x
2
−
12
x
+
11
Now
R
f
′
(
x
)
=
l
i
m
h
→
0
f
(
x
+
h
)
−
f
(
x
)
)
h
=
l
i
m
h
→
0
[
(
x
+
h
)
3
+
6
(
x
+
h
)
2
+
11
(
x
+
h
)
−
6
]
−
[
x
3
−
6
x
2
+
11
x
−
6
]
h
=
l
i
m
h
→
0
(
x
+
h
)
3
−
x
3
−
6
(
x
+
h
)
2
−
x
2
+
11
(
x
+
h
)
−
x
)
h
=
l
i
m
h
→
0
(
x
+
h
)
3
−
x
3
h
−
6
l
i
m
h
→
0
[
(
x
+
h
)
2
−
x
2
]
h
+
11
l
i
m
h
→
0
(
x
+
h
)
−
x
h
=
3
x
2
−
12
x
+
11.
Similarly
L
f
′
(
x
)
=
l
i
m
h
→
0
f
(
x
−
h
)
−
f
(
x
)
−
h
=
l
i
m
h
→
0
(
x
−
h
)
3
−
x
3
−
h
−
6
l
i
m
h
→
0
(
x
−
h
)
2
−
x
2
−
h
+
11
l
i
m
h
→
0
(
x
−
h
)
−
x
−
h
=
3
x
2
−
12
x
+
11
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Similar questions
Q.
Verify Rolle's theorum for the function
f
(
x
)
=
x
3
−
6
x
2
+
11
x
−
6
in the interval
[
1
,
3
]
.
Q.
It is given that for the function
f
(
x
)
=
x
3
−
6
x
2
+
a
x
+
b
Rolle's theorem holds in
[
1
,
3
]
with
c
=
2
+
1
√
3
. Find the value of
′
a
′
and
′
b
′
.
Q.
If the function
f
(
x
)
=
x
3
−
6
x
2
+
a
x
+
b
defined on
[
1
,
3
]
satisfies the rolle's theorem for
c
=
2
√
3
+
1
√
3
then
Q.
If the function
f
(
x
)
=
x
3
−
6
x
2
+
a
x
+
b
defined on [1, 3], satisfies the Rolle's theorem for
c
=
2
√
3
+
1
√
3
, then
Q.
Rolle's theorem is not applicable for the function
f
(
x
)
=
|
x
|
in the interval
[
−
1
,
1
]
because:
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