For reference; a closed geodesic is a geodesic that is a closed loop that is smooth at the origin.
I have been stuck on this for a few days; the only theorem I have relating to even dimension has to do with orientability, so it doesn't seem like there's anything I will be able to do with that. The fact that it is an even dimensional sphere makes it seem like I may have to use the fact that there is no non-vanishing vector field, but I am not sure how to make this work.
Is it possible I can have some hints?
More generally, if $M$ is a Riemannian manifold of sectional curvature $\ge \kappa>0$ then every closed geodesic in $M$ has conjugate points. This is a direct corollary of the Rauch Comparison Theorem, where you compare $M$ with the 2-sphere equipped with the metric of constant curvature $=\kappa$. My favorite reference for the RCT is do Carmo's "Riemannian Geometry", see Thm. 2.3 and Prop. 2.4, Chapter 10.