A particle is travelling from point A to point B. These two points are separated by distance D. Assume that the initial velocity of the particle is zero.

Given that the particle never increases its acceleration along its journey, and that the particle arrives at point B with speed V, what is the longest time that the particle can take to arrive at B?

There can always be a slower accleration. For example, the acceleration could be 2m/s but then slower as 1m/s but then could be made slower as 0.5m/s. Therefore, there is no maximum time it could take because it could always take longer.