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KVPY SX SB Exam 20141 Question Paper
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© examsnet.com
Question : 13
Total: 120
A continuous function
f
:
R
→
R
satisfies the equation
f
(
x
)
=
x
+
x
∫
0
f
(
t
)
d
t
.
Which o f the following options is true?
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
f
(
x
+
y
)
=
f
(
x
)
f
(
y
)
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
f
(
x
)
f
(
y
)
f
(
x
+
y
)
=
f
(
x
y
)
Validate
Solution:
f
(
x
)
=
x
+
x
∫
0
f
(
t
)
d
t
f
′
(
x
)
=
1
+
f
(
x
)
⇒
f
′
(
x
)
−
f
(
x
)
=
1
⇒
e
−
x
f
′
(
x
)
−
f
(
x
)
e
−
x
=
e
−
x
⇒
d
d
x
(
f
(
x
)
e
−
x
)
=
e
−
x
⇒
f
(
x
)
e
−
x
=
e
−
x
−
1
+
c
⇒
f
(
x
)
=
−
1
+
ce
x
f
(
0
)
=
0
=
−
1
+
ce
0
⇒
c
=
1
f
(
x
)
=
e
x
−
1
f
(
x
)
+
f
(
y
)
+
f
(
x
)
f
(
y
)
=
e
x
−
1
+
e
y
−
1
+
(
e
x
−
1
)
(
e
y
−
1
)
=
e
x
−
1
+
e
y
−
1
+
e
x
.
e
y
−
e
y
−
e
x
+
1
=
e
x
.
e
y
−
1
=
e
x
+
y
−
1
=
f
(
x
+
y
)
© examsnet.com
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