Question

Does potential energy have a maximum value when the mass is at rest?

Answer 1

Not necessarily,

Suppose you have kept your book on the table,so it is at rest,and suppose it has an amount of potential energy `mgh` where,`m` is the mass of the book and `h` is the height of the table.

Now,if you put it on another table of height `2h`,its potential energy becomes `2mgh`

So,see on both the tables the book was at rest but their condition at rest didn't let them have maximum potential energy,at both the positions.

Again you can think that while moving the book from one table to the other,if you cause the move with constant velocity `v`,at mid point of its journey,the book had potential energy `mgh+mg(h/2)=(3mgh)/2`,so inspite of being at motion,the book has more potential energy than when it was at rest lying on the `1` st table.