The Attempt at a Solution
I've managed to do parts i) and ii) with not much bother. But as for iii) then I haven't a clue how to show that the period of oscillation is given by that. I've always been under the impression it is simply given by 2pi*sqrt(m/k), but am now wondering whether that was just an approximation in itself. It's clear that the limits would be xb and xa as they define the entire range of the object's motion in the potential curve.
I have a feeling energy considerations may play a part but am not sure how to go about it.