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quadratic, bounding |f'(x)| given bounds on |f(x)|

Source: Bulgaria 1986 P2

June 15, 2021
quadraticsPolynomialsalgebrapolynomial

Problem Statement

Let f(x)f(x) be a quadratic polynomial with two real roots in the interval [−1,1][-1,1]. Prove that if the maximum value of ∣f(x)∣|f(x)| in the interval [−1,1][-1,1] is equal to 11, then the maximum value of ∣f′(x)∣|f'(x)| in the interval [−1,1][-1,1] is not less than 11.
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