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Byju's Answer
Standard XII
Mathematics
Integration as Antiderivative
Consider a co...
Question
Consider a continuous function and differentiable function
′
f
′
satisfies the functional equation
f
(
x
)
+
f
(
y
)
=
f
(
x
+
y
+
x
y
)
+
x
y
for
x
,
y
>
−
1
and
f
′
(
0
)
=
0
,then
The value of
1
f
′′
(
2
)
is
A
9
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B
−
9
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C
4
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D
−
4
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Solution
The correct option is
A
−
9
f
(
x
)
+
f
(
y
)
=
f
(
x
+
y
+
x
y
)
+
x
y
f
(
x
)
+
f
(
y
)
=
f
(
x
+
y
(
1
+
x
)
)
+
x
y
Put x=0 gives f(0) = 0
also given
f
′
(
0
)
=
0
hence
lim
x
→
0
f
(
x
)
x
=
0
f
(
x
+
y
(
1
+
x
)
)
−
f
(
x
)
y
(
1
+
x
)
=
f
(
y
)
−
x
y
y
(
1
+
x
)
Putting limit
y
→
0
l
i
m
y
→
0
f
(
x
+
y
(
1
+
x
)
)
−
f
(
x
)
y
(
1
+
x
)
=
lim
y
→
0
f
(
y
)
y
−
x
(
1
+
x
)
f
′
(
x
)
=
f
′
(
0
)
−
x
(
1
+
x
)
f
′
(
x
)
=
−
x
(
1
+
x
)
Differentiate using product rule
f
′′
(
x
)
=
−
1
(
1
+
x
)
2
f
′′
(
2
)
=
−
1
9
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0
Similar questions
Q.
If
f
is a differentiable function satisfying
f
(
x
y
)
=
f
(
x
)
+
f
(
y
)
+
x
+
y
−
1
x
y
for all
x
,
y
>
0
and
f
′
(
1
)
=
2
,
then the value of
[
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e
100
)
]
is
(where
[
.
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represents the greatest integer function)
Q.
A function
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→
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satisfies the equation
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(
x
+
y
)
=
f
(
x
)
,
f
(
y
)
for all
x
,
y
ϵ
R
,
f
(
x
)
≠
0.
Suppose that the function is differentiable at
x
=
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and
f
′
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0
)
=
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,
then
f
′
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Q.
A function
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⟶
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satisfies the equation
f
(
x
+
y
)
=
f
(
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)
,
f
(
y
)
for all
x
,
y
ϵ
R
,
f
(
x
)
≠
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Suppose that the function is differentiable at x=0 and
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′
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)
=
2
prove that
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′
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Q.
Let
f
be a differentiable function on
R
and satisfies
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
∀
x
,
y
ϵ
R
. If
f
′
(
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)
=
2
, then the value of
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4
)
is
Q.
Let
f
:
R
→
R
be a differentiable function satisfying
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
x
y
for all
x
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y
∈
R
and
lim
h
→
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1
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f
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=
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Which of the following statements is (are) CORRECT?
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