Abstract

Simple necessary and sufficient conditions that a quartic polynomial $f(z)$ be nonnegative for $z > 0$ or $a \leq z \leq b$ are derived, and illustrated geometrically. The geometry provides considerable insight and suggests various approximations and computational simplifications. The theory is applied to monotone quintic spline interpolation, giving necessary and sufficient conditions and an algorithm for monotone Hermite quintic interpolation.

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