Full range of blow up exponents for the quintic wave equation in three dimensions

For the critical focusing wave equation $\square u=u^5$ on $\mathbb{R}^{3+1}$ in the radial case, we prove the existence of type II blow up solutions with scaling parameter $\lambda(t)=t^{-1\nu}$ for all $\nu>0$. This extends the previous work by the authors and Tataru where the condition $\nu>\frac{1}{2}$ had been imposed, and gives the optimal range of polynomial blow up rates in light of recent work by Duyckaerts, Kenig and Merle.