International Journal of Mathematics and Mathematical Sciences
Volume 27 (2001), Issue 2, Pages 125-130
Illumination by Taylor polynomials
Penn State University, 25 Yearsley Mill Road, Media 19063, PA, USA
Received 22 November 1999
Copyright © 2001 Alan Horwitz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let be a differentiable function on the real line
, and let be a point not on the graph of .
Define the illumination index of to be the number of distinct tangents to the graph of which pass through . We prove
that if is continuous and nonnegative on , outside a closed interval of , and
has finitely many zeros on , then any point below the graph of has illumination index . This result fails in general if is not bounded away from on . Also, if has finitely many zeros and is not nonnegative on
, then some point
below the graph has illumination index not equal to . Finally, we generalize our results to illumination by odd order Taylor polynomials.