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KVPY SX SB Exam 2016 Question Paper
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© examsnet.com
Question : 15
Total: 120
The number of continuous function f : [0, 1] → [0, 1] such that
f
(
x
)
<
x
2
for all x and
1
∫
0
f
(
x
)
d
x
=
1
3
is :
0
1
2
infinite
Validate
Solution:
∴ f(x) is always positive for x ∈ [0, 1]
∴
f
(
x
)
<
x
2
⇒
1
∫
0
f
(
x
)
d
x
<
1
∫
0
x
2
I
≤
1
3
But it is given that
I
=
1
3
which is not possible
© examsnet.com
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