Illumination by Taylor polynomials

Alan Horwitz

International Journal of Mathematics and Mathematical Sciences, 2001, vol. 27, 1-6

Abstract:

Let f ( x ) be a differentiable function on the real line ℝ , and let P be a point not on the graph of f ( x ) . Define the illumination index of P to be the number of distinct tangents to the graph of f which pass through P . We prove that if f ″ is continuous and nonnegative on ℝ , f ″ ≥ m > 0 outside a closed interval of ℝ , and f ″ has finitely many zeros on ℝ , then any point P below the graph of f has illumination index 2 . This result fails in general if f ″ is not bounded away from 0 on ℝ . Also, if f ″ has finitely many zeros and f ″ is not nonnegative on ℝ , then some point below the graph has illumination index not equal to 2 . Finally, we generalize our results to illumination by odd order Taylor polynomials.

Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:634045

DOI: 10.1155/S0161171201004173

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