A particle moves along a straight line so that its position (in meters) at time t seconds is given by s(t)=3t3−12t2+9t+5 for 0≤t≤4.
(a) Derive an expression for the particle’s velocity v(t).
(b) Explain why any interior time t at which the particle attains a maximum or minimum position must satisfy v(t)=0, and discuss why the endpoints t=0 and t=4 must also be examined.
(c) Identify all times in [0,4] at which the particle could achieve an extreme position, and classify each as a critical point or an endpoint. Justify your classifications.
(d) Using your results from part (c), determine the greatest distance of the particle from the origin on the interval [0,4], and state the corresponding time.
(e) Interpret the meaning of your answer to part (d) in the context of the particle’s motion, including appropriate units.