Illumination by Taylor Polynomials

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 thru 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 zeroes on R, then every point 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 zeroes and f '' is not nonnnegative 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.