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KEAM 2018 Math Paper
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© examsnet.com
Question : 15
Total: 120
Let
f
:
R
→
R
satisfy
f
(
x
)
f
(
y
)
=
f
(
x
y
)
for all real numbers x and y. If
f
(
2
)
=
4
,
then
f
′
(
1
2
)
=
0
1
4
1
2
1
2
Validate
Solution:
Given,
f
:
∙
→
∙
f
(
x
)
f
(
y
)
=
f
(
x
y
)
......(i)
On taking
x
=
1
,
y
=
1
f
(
1
)
f
(
1
)
=
f
(
1
⋅
1
)
=
f
(
1
)
2
=
f
(
1
)
=
f
(
1
)
=
1
Now,
x
=
2
,
y
=
1
2
,
then from Eq. (i)
f
(
2
)
f
(
1
2
)
=
f
(
2
⋅
1
2
)
⇒
4
⋅
f
(
1
2
)
=
f
(
1
)
[
∵
f
(
2
)
=
4
]
⇒
f
(
1
2
)
=
1
4
f
(
1
)
On putting the value of
f
(
1
)
⇒
f
(
1
2
)
=
1
4
⋅
1
=
1
4
© examsnet.com
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