Existence of vector extensions for the Hessian

Question

Q: $p$ a critical point of a smooth $f$ and $v, w$ two vectors in $T_{p}M$. We extend these two to vector fields $v^{*}, w^{*}$ such that at $p$ the first one equals $v$ and the second one equals $w.$ Then the Hessian $d_{p}^{2}f(v,w) = L_{v^{*}}L_{w^{*}}f.$ NTS: a) that $v^{*}, w^{*}$ exist and the Hessian is independent of this choice. Here i also need to state the importance of $p$ being a critical point. b)Want to prove that the Hessian is greater or equal to 0 for all $v$ when $p$ is a local min of $f.$ And $p$ is a local min when the Hessian is greater or equal to $0$ for all non 0 $v.$