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Vojtěch Jarník IMC
1997 VJIMC
Problem 3
f^(k)(0)>0 for k=1,2,...
f^(k)(0)>0 for k=1,2,...
Source: VJIMC 1997 1.3
September 30, 2021
calculus
differentiation
Problem Statement
Let
c
1
,
c
2
,
…
,
c
n
c_1,c_2,\ldots,c_n
c
1
,
c
2
,
…
,
c
n
be real numbers such that
c
1
k
+
c
2
k
+
…
+
c
n
k
>
0
for all
k
=
1
,
2
,
…
c_1^k+c_2^k+\ldots+c_n^k>0\qquad\text{for all }k=1,2,\ldots
c
1
k
+
c
2
k
+
…
+
c
n
k
>
0
for all
k
=
1
,
2
,
…
Let us put
f
(
x
)
=
1
(
1
−
c
1
x
)
(
1
−
c
2
x
)
⋯
(
1
−
c
n
x
)
.
f(x)=\frac1{(1-c_1x)(1-c_2x)\cdots(1-c_nx)}.
f
(
x
)
=
(
1
−
c
1
x
)
(
1
−
c
2
x
)
⋯
(
1
−
c
n
x
)
1
.
z
∈
C
z\in\mathbb C
z
∈
C
Show that
f
(
k
)
(
0
)
>
0
f^{(k)}(0)>0
f
(
k
)
(
0
)
>
0
for all
k
=
1
,
2
,
…
k=1,2,\ldots
k
=
1
,
2
,
…
.
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