MathDB
closed bounded interval, non-constant monic polynomial function, max inequality

Source: IMAR 2015 p4

September 27, 2018
functionintervalpolynomialmonicanalysisreal analysisalgebra

Problem Statement

(a) Show that, if I⊂RI \subset R is a closed bounded interval, and f:I→Rf : I \to R is a non-constant monic polynomial function such that maxx∈I∣f(x)∣<2max_{x\in I}|f(x)|< 2, then there exists a non-constant monic polynomial function g:I→Rg : I \to R such that maxx∈I∣g(x)∣<1max_{x\in I} |g(x)| < 1. (b) Show that there exists a closed bounded interval I⊂RI \subset R such that maxx∈I∣f(x)∣≥2max_{x\in I}|f(x)| \ge 2 for every non-constant monic polynomial function f:I→Rf : I \to R.
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