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Derivative and bounds of a monic polynomial

Source: Mexican University Math Olympiad 2024, Problem 6

October 1, 2024
polynomialcalculusderivativeboundedalgebra

Problem Statement

Let p p be a monic polynomial with all distinct real roots. Show that there exists K K such that (p(x)2)′′≤K(p′(x))2. (p(x)^2)'' \leq K(p'(x))^2.
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