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National and Regional Contests
India Contests
ISI Entrance Examination
2014 ISI Entrance Examination
4
Identically zero
Identically zero
Source: ISI Entrance 2014, P4
May 11, 2014
trigonometry
real analysis
Problem Statement
Let
f
,
g
f,g
f
,
g
are defined in
(
a
,
b
)
(a,b)
(
a
,
b
)
such that
f
(
x
)
,
g
(
x
)
∈
C
2
f(x),g(x)\in\mathcal{C}^2
f
(
x
)
,
g
(
x
)
∈
C
2
and non-decreasing in an interval
(
a
,
b
)
(a,b)
(
a
,
b
)
. Also suppose
f
′
′
(
x
)
=
g
(
x
)
,
g
′
′
(
x
)
=
f
(
x
)
f^{\prime \prime}(x)=g(x),g^{\prime \prime}(x)=f(x)
f
′′
(
x
)
=
g
(
x
)
,
g
′′
(
x
)
=
f
(
x
)
. Also it is given that
f
(
x
)
g
(
x
)
f(x)g(x)
f
(
x
)
g
(
x
)
is linear in
(
a
,
b
)
(a,b)
(
a
,
b
)
. Show that
f
≡
0
and
g
≡
0
f\equiv 0 \text{ and } g\equiv 0
f
≡
0
and
g
≡
0
in
(
a
,
b
)
(a,b)
(
a
,
b
)
.
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