Assuming that the professor ends his/her trip at 0 m/s. (Isupposehe could crash his car at the end of the problem, but then we can calculate the deceleration at the end and find out it is enormously larger than the given answer. ) So the least possible acceleration will be given by a constant acceleration to the half-way point and the same magnitude constant deceleration to the destination. (That this gives the "minimum possible maximum" acceleration is guaranteed by the MVT.)

So the acceleration may be calcuated by

The professor travels S/2 m to get to the half-way point, in a time T/2.

Thus

Solve for a. This is the least possible maximum acceleration to get to the destination on time.

-Dan