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Sweden Contests
Swedish Mathematical Competition
1970 Swedish Mathematical Competition
4
4
Part of
1970 Swedish Mathematical Competition
Problems
(1)
p(x) p''(x) <= p'(x)^2
Source: 1970 Swedish Mathematical Competition p4
3/21/2021
Let
p
(
x
)
=
(
x
−
x
1
)
(
x
−
x
2
)
(
x
−
x
3
)
p(x) = (x- x_1)(x- x_2)(x- x_3)
p
(
x
)
=
(
x
−
x
1
)
(
x
−
x
2
)
(
x
−
x
3
)
, where
x
1
,
x
2
x_1, x_2
x
1
,
x
2
and
x
3
x_3
x
3
are real. Show that
p
(
x
)
p
′
′
(
x
)
≤
p
′
(
x
)
2
p(x) p''(x) \le p'(x)^2
p
(
x
)
p
′′
(
x
)
≤
p
′
(
x
)
2
for all
x
x
x
.
Cubic
analysis
algebra
polynomial