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f(x)f(x + 1) < 0, f(x)f(x -1) < 0 for f(x) = ax^2 + bx + c

Source: 2005 Cuba MO 2.2

September 15, 2024
algebratrinomialinequalities

Problem Statement

Determine the quadratic functions f(x)=ax2+bx+cf(x) = ax^2 + bx + c for which there exists an interval (h,k)(h, k) such that for all x∈(h,k)x \in (h, k) it holds that f(x)f(x+1)<0f(x)f(x + 1) < 0 and f(x)f(x−1)<0f(x)f(x -1) < 0.
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