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Illumination by Taylor polynomials

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  • Alan Horwitz

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.

Suggested Citation

  • Alan Horwitz, 2001. "Illumination by Taylor polynomials," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 27, pages 1-6, January.
    • Handle: RePEc:hin:jijmms:634045
    DOI: 10.1155/S0161171201004173
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