First off, find the range of mechanical energies. What could it possibly be if the motion is bounded? What does "bounded" mean classically?
Is it -u<E<0 ? The motion of the system is bounded if it begins at a point situated between two forbidden regions, the system’s trajectory will never leave the allowed region where it started (otherwise it would have to cross one of the forbidden regions).
No, I can tell you it is not true.
U is the potential energy
E is the total energy of the particle, an independent variable in this exercise.
As you almost say (well, guess) in #3, the given information tthat the particle is bounded leads to ##-u \le E < 0##.
With K the kinetic energy and, as you say, E = K + U you can write down the bounds of K and thereby the bounds of v. And derive the period of the motion.