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f(x) = ax^3 + bx^2 + cx + d, |f(x)| <= 1 for all -1 <= x <= 1

Source: 1965 Swedish Mathematical Competition p5

March 21, 2021
Cubicalgebrapolynomialbounded

Problem Statement

Let SS be the set of all real polynomials f(x)=ax3+bx2+cx+df(x) = ax^3 + bx^2 + cx + d such that ∣f(x)∣≤1|f(x)| \le 1 for all −1≤x≤1 -1 \le x \le 1. Show that the set of possible ∣a∣|a| for ff in SS is bounded above and find the smallest upper bound.
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