Let $(M,g)$ be a Riemannian manifold, and let $\gamma:[0,R]\to M$ be a unit-speed geodesic segment whose radial sectional curvatures satisfy $\operatorname{sec}\le k$. Let $J$ be a Jacobi field along $\gamma$ such that $J(0)=0$, $D_tJ(0)=E_0$, $|E_0|=1$, and $J(t)\perp \dot{\gamma}(t)$ for $0<t\le R$. Assume $\gamma|_{[0,R]}$ has no conjugate point to $\gamma(0)$, and let $\operatorname{sn}_k$ be the model solution of $s''+ks=0$, $s(0)=0$, $s'(0)=1$. Rauch comparison gives $|J(t)|\ge \operatorname{sn}_k(t)$ for all $t\in(0,R]$ for which $\operatorname{sn}_k(t)>0$. If $|J(t_0)|=\operatorname{sn}_k(t_0)$ for some $t_0\in(0,R]$, then for every $t\in[0,t_0]$ the field has the form $J(t)=\operatorname{sn}_k(t)E(t)$ for a parallel unit field $E(t)$ perpendicular to $\dot{\gamma}(t)$, and $\operatorname{sec}(\dot{\gamma}(t),E(t))=k$ wherever $t>0$.