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Roots of Real valued function

Source: LIMIT 2020 Cat 2 Obj P6

May 25, 2020
limitfunctionsalgebrafunction

Problem Statement

Let f(x)f(x) be a real-valued function satisfying af(x)+bf(−x)=px2+qx+raf(x)+bf(-x)=px^2+qx+r. aa and bb are distinct real numbers and p,q,rp,q,r are non-zero real numbers. Then f(x)=0f(x)=0 will have real solutions when
(A)(a+ba−b)≤q24pr\left(\frac{a+b}{a-b}\right)\leq\frac{q^2}{4pr} (B)(a+ba−b)≤4prq2\left(\frac{a+b}{a-b}\right)\leq\frac{4pr}{q^2} (C)(a+ba−b)≥q24pr\left(\frac{a+b}{a-b}\right)\geq\frac{q^2}{4pr} (D)(a+ba−b)≥4prq2\left(\frac{a+b}{a-b}\right)\geq\frac{4pr}{q^2}
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