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ISI B.Stat Entrance Exam
2011 ISI B.Stat Entrance Exam
4
4
Part of
2011 ISI B.Stat Entrance Exam
Problems
(1)
f(0)=1, f(x) \ge 0 \ge f'(x), f"(x)\le f'(x) for x\ge 0
Source: ISI BS 2011 P4
3/31/2013
Let
f
f
f
be a twice differentiable function on the open interval
(
−
1
,
1
)
(-1,1)
(
−
1
,
1
)
such that
f
(
0
)
=
1
f(0)=1
f
(
0
)
=
1
. Suppose
f
f
f
also satisfies
f
(
x
)
≥
0
,
f
′
(
x
)
≤
0
f(x) \ge 0, f'(x) \le 0
f
(
x
)
≥
0
,
f
′
(
x
)
≤
0
and
f
′
′
(
x
)
≤
f
(
x
)
f''(x) \le f(x)
f
′′
(
x
)
≤
f
(
x
)
, for all
x
≥
0
x\ge 0
x
≥
0
. Show that
f
′
(
0
)
≥
−
2
f'(0) \ge -\sqrt2
f
′
(
0
)
≥
−
2
.
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