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Problems
If
f
(
x
)
=
lim
n
→
∞
(
sin
4
x
+
1
4
sin
4
2
x
+
⋯
+
1
4
n
⋅
sin
4
(
2
n
x
)
)
f(x) = \lim_{n \to \infty}\left(\sin^4 x + \frac{1}{4}\sin^4 2x + \dots + \frac{1}{4^n}\cdot \sin^4(2^n x)\right)
f
(
x
)
=
lim
n
→
∞
(
sin
4
x
+
4
1
sin
4
2
x
+
⋯
+
4
n
1
⋅
sin
4
(
2
n
x
)
)
…
Problem
If
f
(
x
)
=
lim
n
→
∞
(
sin
4
x
+
1
4
sin
4
2
x
+
⋯
+
1
4
n
⋅
sin
4
(
2
n
x
)
)
f(x) = \lim_{n \to \infty}\left(\sin^4 x + \frac{1}{4}\sin^4 2x + \dots + \frac{1}{4^n}\cdot \sin^4(2^n x)\right)
f
(
x
)
=
lim
n
→
∞
(
sin
4
x
+
4
1
sin
4
2
x
+
⋯
+
4
n
1
⋅
sin
4
(
2
n
x
)
)
and
g
(
x
)
g(x)
g
(
x
)
is a differentiable function satisfying
g
(
x
)
+
f
(
x
)
=
1
g(x) + f(x) = 1
g
(
x
)
+
f
(
x
)
=
1
, then the maximum value of
(
f
(
x
)
+
g
(
x
)
)
4
(\sqrt{f(x)} + \sqrt{g(x)})^4
(
f
(
x
)
+
g
(
x
)
)
4
is .....
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