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National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1991 Bulgaria National Olympiad
Problem 4
Problem 4
Part of
1991 Bulgaria National Olympiad
Problems
(1)
P(x)P''(x)≤P'(x)^2 if P has deg P roots
Source: Bulgaria 1991 P4
6/3/2021
Let
f
(
x
)
f(x)
f
(
x
)
be a polynomial of degree
n
n
n
with real coefficients, having
n
n
n
(not necessarily distinct) real roots. Prove that for all real
x
x
x
,
f
(
x
)
f
′
′
(
x
)
≤
f
′
(
x
)
2
.
f(x)f''(x)\le f'(x)^2.
f
(
x
)
f
′′
(
x
)
≤
f
′
(
x
)
2
.
algebra
Polynomials