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Putnam 1941 A4

Source: Putnam 1941

February 23, 2022
Putnampolynomialroots

Problem Statement

Let the roots a,b,ca,b,c of f(x)=x3+px2+qx+rf(x)=x^3 +p x^2 + qx+r be real, and let a≤b≤ca\leq b\leq c. Prove that f′(x)f'(x) has a root in the interval [b+c2,b+2c3]\left[\frac{b+c}{2}, \frac{b+2c}{3}\right]. What will be the form of f(x)f(x) if the root in question falls at either end of the interval?
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