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Mathematics
Mathematics
Problem
If
f
f
f
is a differentiable function satisfying
f
(
1
n
)
=
0
,
∀
n
≥
1
,
n
∈
I
f\left(\frac{1}{n}\right) = 0, \forall n \ge 1, n \in \mathbb{I}
f
(
n
1
)
=
0
,
∀
n
≥
1
,
n
∈
I
, then a.
f
(
x
)
=
0
,
x
∈
(
0
,
1
]
f(x) = 0, x \in (0, 1]
f
(
x
)
=
0
,
x
∈
(
0
,
1
]
b.
f
′
(
0
)
=
0
=
f
(
0
)
f'(0) = 0 = f(0)
f
′
(
0
)
=
0
=
f
(
0
)
c.
f
(
0
)
=
0
f(0) = 0
f
(
0
)
=
0
but
f
′
(
0
)
f'(0)
f
′
(
0
)
not necessarily zero d.
∣
f
(
x
)
∣
≤
1
,
x
∈
(
0
,
1
]
|f(x)| \le 1, x \in (0, 1]
∣
f
(
x
)
∣
≤
1
,
x
∈
(
0
,
1
]
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