Does a Lyapunov Function derivative necessarily vanish at each equilibrium point of the system?
Lyapunov functions are a tool used to demonstrate stability of systems. Imagine that we are interested only in demonstrating the stability of say, five equilibrium points out of say, total ten equilibrium points of a system. Then, we need to search for candidate Lyapunov functions which have maxima/minima at the  equilibrium points of interest only. Those candidate functions may or may not have maxima/minima at other equilibrium points. Proof of local stability about any given (isolated) equilibrium point doesn't depend on the behaviour of the selected Lyapunov function at other equilibrium points.