Illumination by Taylor Polynomials
نویسنده
چکیده
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 onR, then any point P below the graph of f has illumination index 2. This result fails in general if f ′′ is not bounded away from0 onR. Also, if f ′′ has finitelymany 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. 2000 Mathematics Subject Classification. 26A06.
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