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Mathematics
Mathematics
Problem
Let
f
:
[
0
,
1
]
→
R
f: [0, 1] \to \mathbb{R}
f
:
[
0
,
1
]
→
R
be such that
f
(
x
y
)
=
f
(
x
)
f
(
y
)
f(xy) = f(x)f(y)
f
(
x
y
)
=
f
(
x
)
f
(
y
)
for all
x
,
y
∈
[
0
,
1
]
x, y \in [0, 1]
x
,
y
∈
[
0
,
1
]
, and
f
(
0
)
≠
0
f(0) \neq 0
f
(
0
)
=
0
. If
y
=
y
(
x
)
y = y(x)
y
=
y
(
x
)
satisfies the differential equation,
d
y
d
x
=
f
(
x
)
\frac{dy}{dx} = f(x)
d
x
d
y
=
f
(
x
)
with
y
(
0
)
=
1
y(0) = 1
y
(
0
)
=
1
, then
y
(
1
4
)
+
y
(
3
4
)
y\left(\frac{1}{4}\right) + y\left(\frac{3}{4}\right)
y
(
4
1
)
+
y
(
4
3
)
is equal to : [JEE-MAIN ONLINE-2019] (1) 4 (2) 3 (3) 5 (4) 2
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