MathDB

Cubic polynomial with absolute value inequality

Source: IMO Shortlist 1996, A5

August 9, 2008

algebrapolynomialinequalitiesfunctionmaximizationIMO Shortlist

Problem Statement

Let

P(x)

be the real polynomial function, P(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d. Prove that if

|P(x)| \leq 1

for all

x

such that

|x| \leq 1,

then |a| \plus{} |b| \plus{} |c| \plus{} |d| \leq 7.