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Byju's Answer
Standard XII
Mathematics
Algebra of Derivatives
Given that ...
Question
Given that
f
(
x
)
=
x
g
(
x
)
|
x
|
,
g
(
0
)
=
g
′
(
0
)
=
0
and
f
is continuous at
x
=
0
, the value of
f
′
(
0
)
is
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Solution
f
(
x
)
=
x
g
(
x
)
|
x
|
⇒
f
(
x
)
=
{
g
(
x
)
x
≥
0
−
g
(
x
)
x
<
0
f
(
x
)
=
lim
h
→
0
g
(
x
+
h
)
−
g
(
x
)
x
+
h
−
x
⇒
f
′
(
0
)
=
lim
h
→
0
+
g
(
0
+
h
)
−
g
(
0
)
h
=
lim
h
→
0
g
(
h
)
h
=
g
′
(
0
)
=
0
Hence, correcr answer is
0
.
Suggest Corrections
0
Similar questions
Q.
Assertion :
Let f & g be real valued functions defined on interval (-1, 1) such that
g
′′
(
x
)
is continuous,
g
(
0
)
≠
0
,
g
′
(
0
)
=
0
,
g
n
(
0
)
≠
0
&
f
(
x
)
=
g
(
x
)
sin
x
.
lim
x
→
0
[
g
(
x
)
cot
x
−
g
(
0
)
c
o
s
e
c
x
]
=
f
′′
(
0
)
Reason:
f
′
(
0
)
=
g
(
0
)
Q.
Assertion :Let
f
and
g
be real-valued functions defined on interval
(
−
1
,
1
)
such that
g
′′
(
x
)
is countinuous,
g
(
0
)
≠
0
,
g
′
(
0
)
=
0
,
g
′′
(
0
)
≠
0
and
f
(
x
)
=
g
(
x
)
.
sin
x
.
lim
x
→
0
{
g
(
x
)
⋅
cot
x
−
g
(
0
)
⋅
c
o
sec
x
=
f
′′
(
0
)
}
,
and
Reason:
f
′
(
0
)
=
g
(
0
)
.
Q.
Let
f
and
g
be real valued functions defined on interval
(
−
1
,
1
)
such that
g
′′
(
x
)
is continuous,
g
(
0
)
≠
0
,
g
′
(
0
)
=
0
,
g
′′
(
0
)
≠
0
, and
f
(
x
)
=
g
(
x
)
sin
x
.
STATEMENT -1 :
lim
x
→
0
[
g
(
x
)
cot
x
−
g
(
0
)
c
o
s
e
c
x
]
=
f
′′
(
0
)
STATEMENT-2:
f
′
(
0
)
=
g
(
0
)
.
Q.
If
f
is differentiable function satisfying
f
(
0
)
=
0
and if
g
(
x
)
=
f
(
x
)
x
, then value, that should be assigned to
g
(
0
)
, so that
g
is continuous at
0
is
Q.
The value of
f
at
x
=
0
so that function
f
(
x
)
=
2
x
−
2
−
x
x
,
x
≠
0
, is continuous at
x
=
0
, is
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