The potential is zero at the origin, so the particle will have kinetic energy equal to there. So, its velocity will be . If the particle is moving to the right, then its acceleration will be . The time it will take for the particle to come to rest is . Then the particle will start moving backward toward the origin. The time it will take to do that is also . So, the total time the particle spends on the right side of the -axis is .
Now, assume the particle is at the origin and is moving to the left with velocity . If the potential were for (instead of only ), would also be at the origin and moving to the left with speed at some point in time during its period. The period for the potential for is
By symmetry, the particle spends half of its time to the left of the y-axis and half of its time to the right of the y-axis. So, the amount of time it spends to the left of the axis is
We add the amount of time the particle spends to the left of the y-axis to the amount of time the particle spends to the right of the -axis to obtain the period:
Therefore, answer (D) is correct.