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Maximum Value of a Polynomial

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October 22, 2010
algebrapolynomialalgebra unsolved

Problem Statement

A polynomial P(x)P (x) with real coefficients and of degree n≥3n \ge 3 has nn real roots x1<x2<⋯<xnx_1 <x_2 < \cdots < x_n such that x2−x1<x3−x2<⋯<xn−xn−1x_2 - x_1 < x_3 - x_2 < \cdots < x_n - x_{n-1} Prove that the maximum value of ∣P(x)∣|P (x)| on the interval [x1,xn][x_1 , x_n ] is attained in the interval [xn−1,xn][x_{n-1} , x_n ].
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