Illumination by Tangent Lines

TL;DR

This paper investigates the illumination index of points relative to the graph of differentiable functions, establishing conditions under which points are illuminated by one, two, or no tangent lines, with specific results for convex rational functions, exponential polynomials, and polynomials.

Contribution

It provides new theoretical results characterizing the number of tangent lines passing through points outside the graph of certain classes of functions, including convex rational functions and polynomials.

Findings

Points below convex rational functions have illumination index 2.

Conditions for points to have 1 or 2 tangent lines passing through them.

Results on the illumination index for polynomials and functions with specific asymptotes.

Abstract

Let f be a differentiable function on the real line, and let P\inG_{f}^{C}= all points not on the graph of f. We say that the illumination index of P, denoted by I_{f}(P), is k if there are k distinct tangents to the graph of f which pass through P. In section 2 we prove results about the illumination index of f with f" (x)\geq 0 on \Re. In particular, suppose that y=L_1(x) and y=L_2(x) are distinct oblique asymptotes of f and let P=(s,t)\in G_{f}^{C}. If max(L_1(s),L_2(s))<t<f(s), then I_{f}(P)=2. If L_1(s)\not= L_2(s) and min(L_1(s),L_1(s))<t\leqmax(L_1(s),L_2(s)), then I_{f}(P)=1. Finally, if t_\leqmin(L_1(s),L_2(s)), then I_{f}(P)=0. We also show that any point below the graph of a convex rational function or exponential polynomial must have illumination index equal to 2. In section 3 we also prove results about the illumination index of polynomials.