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Let p(x) be a polynomial over R of even degree n for which for which p(x)≥0 for all x. Prove that p(x) + p'(x) +...+ p(n)(x)≥0 for all x.
source :Polynomials, E.J.Barbeau
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21Let q(x) = p(x) + p'(x) +...+ p(n)(x).Then q(x) is a polynomial of even degree with positive leading coefficient. Suppose q(x) assumes its minimum value when x = u. Then 0 = q'(u) = q(u) - p(u). But then q(x)≥q(u)= p(u) ≥ 0
source: Polynomials E.J.Barbeau