The main result of this paper is the following. Let \(I=[0,1]\), and let \(f\in C^ 0(I,I)\) be an expanding map with expanding constant \(\lambda >1\), which is increasing on [0,c] and decreasing on [c,1]. If \([(1+\sqrt{5})/2]^{1/2^ n}\leq \lambda\) for some nonnegative integer n, then f has a periodic point of least period \(2^ m\cdot 3\) in [c,1] for some nonnegative integer \(m\leq n\). The result is extended to certain mappings which are not necessarily expanding. The lower bound for \(\lambda\) is sharp. The results are used to construct one-parameter families of maps on I which have a bifurcation from fixed points directly to periodic points of least period \(2^ k\cdot 3\) for some \(k\geq 0\); i.e. there is a direct bifurcation from simple dynamics (with topological entropy zero) to chaos (with positive topological entropy).