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Byju's Answer
Standard XII
Mathematics
Continuity in an Interval
A function ...
Question
A function
f
:
R
⟶
R
satisfies the equation
f
(
x
+
y
)
=
f
(
x
)
,
f
(
y
)
for all
x
,
y
ϵ
R
,
f
(
x
)
≠
0
Suppose that the function is differentiable at x=0 and
f
′
(
0
)
=
2
prove that
f
′
(
x
)
=
2
f
(
x
)
Open in App
Solution
f
′
(
x
)
=
lim
x
→
0
f
(
x
+
b
)
−
f
(
x
)
h
=
lim
x
→
0
f
(
x
)
f
(
h
)
−
f
(
x
)
h
⇒
lim
x
→
0
f
(
x
)
[
f
(
x
)
−
1
h
]
=
lim
x
→
0
f
(
x
)
[
f
(
h
)
−
f
(
0
)
h
]
∵
f
(
x
+
0
)
=
f
(
x
)
×
f
(
0
)
⇒
f
(
0
)
=
1
f
′
(
x
)
=
f
(
x
)
f
′
(
0
)
f
′
(
x
)
=
2
f
(
x
)
∵
f
′
(
x
)
=
2
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Similar questions
Q.
A function
f
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satisfies the equation
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for all
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Suppose
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)
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=
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