Suppose that the n times differentiable real function f(x) has at least n+1 distinct zeros in the closed interval [a,b] and that the polynomial P(z)=zn+cn−1zn−1+…+c1x+c0 has only real zeroes. Show that
f(n)(x)+cn−1f(n−1)(x)+…+c1f′(x)+c0f(x) has at least one zero in [a,b], where f(n) denotes the n-th derivative of f. Putnampolynomialcalculusderivative