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Putnam 1956 A4

Source: Putnam 1956

July 5, 2022
Putnampolynomialcalculusderivative

Problem Statement

Suppose that the nn times differentiable real function f(x)f(x) has at least n+1n+1 distinct zeros in the closed interval [a,b][a,b] and that the polynomial P(z)=zn+cn−1zn−1+…+c1x+c0P(z)=z^n +c_{n-1}z^{n-1}+\ldots+c_1 x +c_0 has only real zeroes. Show that f(n)(x)+cn−1f(n−1)(x)+…+c1f′(x)+c0f(x)f^{(n)}(x)+ c_{n-1} f^{(n-1)}(x) +\ldots +c_1 f'(x)+ c_0 f(x) has at least one zero in [a,b][a,b], where f(n)f^{(n)} denotes the nn-th derivative of f.f.
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