MathDB
Problems
Contests
National and Regional Contests
India Contests
ISI Entrance Examination
2018 ISI Entrance Examination
4
4
Part of
2018 ISI Entrance Examination
Problems
(1)
ISI 2018 #4
Source: ISI 2018 B.Stat / B.Math Entrance Exam
5/13/2018
Let
f
:
(
0
,
∞
)
→
R
f:(0,\infty)\to\mathbb{R}
f
:
(
0
,
∞
)
→
R
be a continuous function such that for all
x
∈
(
0
,
∞
)
x\in(0,\infty)
x
∈
(
0
,
∞
)
,
f
(
2
x
)
=
f
(
x
)
f(2x)=f(x)
f
(
2
x
)
=
f
(
x
)
Show that the function
g
g
g
defined by the equation
g
(
x
)
=
∫
x
2
x
f
(
t
)
d
t
t
for
x
>
0
g(x)=\int_{x}^{2x} f(t)\frac{dt}{t}~~\text{for}~x>0
g
(
x
)
=
∫
x
2
x
f
(
t
)
t
d
t
for
x
>
0
is a constant function.
isi
2018
Indian Statistical Institute
calculus