Illumination by Taylor polynomials (Q1599737)

Summary: Let \(f(x)\) be a differentiable function on the real line \(\mathbb{R}\), 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 \(\mathbb{R}\), \(f''\geq m>0\) outside a closed interval of \(\mathbb{R}\), and \(f''\) has finitely many zeros on \(\mathbb{R}\), then any point \(P\) \textit{below} the graph of \(f\) has illumination index 2. This result fails in general if \(f''\) is not bounded away from \(0\) on \(\mathbb{R}\). Also, if \(f''\) has finitely many zeros and \(f''\) is not nonnegative on \(\mathbb{R}\), 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.