I had a couple of related questions on the cut locus, conjugate points and smoothness of distance function. Let $(M,g)$ be a smooth complete Riemannian manifold and $r(x) = d(p,x)$ the distance function to a fixed point $p\in M$. Recall that the cut locus $Cut(p)$ consists of two kinds of points - points that are conjugate to $p$ along some geodesic, or points that have multiple minimal geodesics connecting them to $p$.

1) I know that $r$ is smooth on $M\backslash (Cut(p)\cup \{p\}$. Is it true that $r$ is in fact smooth at $x$ as long as there is a unique minimal geodesic connecting it to $p$? The only proof of smoothness involves the exponential map, but the exponential map is not invertible at conjugate points, so that approach wont work.

2) Is there a simple example of $(M,g,p)$ with points conjugate to $p$ but with unique minimal geodesics to $p$.