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Byju's Answer
Standard XII
Mathematics
Theorems for Differentiability
Let f x + y =...
Question
Let
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
and
f
(
x
)
=
x
2
g
(
x
)
for all
x
,
y
ϵ
R
, where
g
(
x
)
is continuous function. Then
f
′
(
x
)
is equal to
A
g'(x)
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B
g(0)
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C
g(0) + g'(x)
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Solution
We have
f
′
(
x
)
=
l
i
m
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
=
l
i
m
h
→
0
f
(
x
)
+
f
(
h
)
−
f
(
x
)
h
[
∵
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
]
=
l
i
m
h
→
0
f
(
h
)
h
=
l
i
m
h
→
0
h
2
g
(
h
)
h
=
0.
g
(
0
)
=
0
[because
g
is continuous therefore
l
i
m
h
→
0
g
(
h
)
=
g
(
0
)
]
.
Suggest Corrections
0
Similar questions
Q.
Let
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
and
f
(
x
)
=
x
2
g
(
x
)
for all
x
,
y
ϵ
R
, where
g
(
x
)
is continuous function. Then
f
′
(
x
)
is equal to
Q.
Let f
be a function such that
f(x+y)=f(x)+f(y)
for all
x and y and
f
(
x
)
=
(
2
x
2
+
3
x
)
g
(
x
)
for all x where g(x) is continuous and g(0)=9 then
f
′
(
x
)
is equals to
Q.
Let f be a function such that f(x+y)=f(x)+f(y) for all x and y and
f
(
x
)
=
(
2
x
2
+
3
x
)
g
(
x
)
for all x where g(x) is continuous and g(0)=9 then f'(0) is equals to
Q.
Given that
f
′
(
x
)
>
g
′
(
x
)
for all real
x
, and
f
(
0
)
=
g
(
0
)
, then
f
(
x
)
<
g
(
x
)
for all
x
belongs to
Q.
Given that
f
′
(
x
)
>
g
′
(
x
)
for all real
x
,
and
f
(
0
)
=
g
(
0
)
,
then
f
(
x
)
<
g
(
x
)
for all
n
belonging to
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