Conditional covariance martingale and increasing function

Question

Consider a discrete time martingale $X$ and a non-random $T>0$. Suppose that $X_T\neq X_{T-1}$ almost surely. Why is it true that $$ \operatorname{Cov}(f(X_T),X_T|F_{T-1})=E[f(X_T)X_T|F_{T-1}]-E[f(X_T)|F_{T-1}]X_{T-1}>0 $$ almost surely for all strictly increasing functions $f$ (for which we have the required integrability)?