12. A point moves along a number line so that its position at time t >= 0 is s(t) = 2t³ – 15t² + 36t – 10. What is the position of the point when it first changes direction?

First find the critical points by solving s’(t) = 0.

Let the two critical points divide the domain of definition into three subintervals and determine the sign of s’(t) within each interval. A change in sign implies a change in direction at the corresponding critical point. As demonstrated by the following graph:

Thus, the position of the point when it first changes direction is given by, solving s(2) as follows: