Let J be a Jacobi field along a geodesic c(t) in a Riemannian manifold (M,g). We define a new function f_J(t) (related in some sense to the norm of J) and show that if curvature of $M$ is (positive) non-negative then f_J must be (strictly) concave. This can be used to prove new obstructions to the existence of positively curved invariant metrics on some cohomogeneity one manifolds. This is a joint work with Wolfgang Ziller.