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Mathematics
Mathematics
Problem
Find the area of the region bounded by
y
=
f
(
x
)
,
y
=
∣
g
(
x
)
∣
y = f(x), y = |g(x)|
y
=
f
(
x
)
,
y
=
∣
g
(
x
)
∣
and the lines
x
=
0
,
x
=
2
x = 0, x = 2
x
=
0
,
x
=
2
, where
f
,
g
f, g
f
,
g
are continuous functions satisfying
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
−
8
x
y
,
∀
x
,
y
∈
R
f(x+y) = f(x) + f(y) - 8xy, \forall x, y \in \mathbb{R}
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
−
8
x
y
,
∀
x
,
y
∈
R
and
g
(
x
+
y
)
=
g
(
x
)
+
g
(
y
)
+
3
x
y
(
x
+
y
)
,
∀
x
,
y
∈
R
g(x+y) = g(x) + g(y) + 3xy(x+y), \forall x, y \in \mathbb{R}
g
(
x
+
y
)
=
g
(
x
)
+
g
(
y
)
+
3
x
y
(
x
+
y
)
,
∀
x
,
y
∈
R
Also,
f
′
(
0
)
=
8
f'(0) = 8
f
′
(
0
)
=
8
and
g
′
(
0
)
=
−
4
g'(0) = -4
g
′
(
0
)
=
−
4
.
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