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Properties of derivatives

Source: Moldovan MO 2003

February 21, 2006
calculusderivativealgebrapolynomialalgebra proposed

Problem Statement

For every natural number n≥2n\geq{2} consider the following affirmation PnP_n: "Consider a polynomial P(X)P(X) (of degree nn) with real coefficients. If its derivative P′(X)P'(X) has n−1n-1 distinct real roots, then there is a real number CC such that the equation P(x)=CP(x)=C has nn real,distinct roots." Are P4P_4 and P5P_5 both true? Justify your answer.
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