AMC 12 2014b Problem 13

Real numbers $a$ and $b$ are chosen with $1<a<b$ such that no triangles with positive area has side lengths $1, a,$ and $b$ or $\tfrac{1}{b}, \tfrac{1}{a},$ and $1$. What is the smallest possible value of $b$?

\textbf{(A)}\ \frac{3+\sqrt{3}}{2} \textbf{(B)}\ \frac{5}{2} \textbf{(C)}\ \frac{3+\sqrt{5}}{2} \textbf{(D)}\ \frac{3+\sqrt{6}}{2} \textbf{(E)}\ 3