A toy rocket is launched vertically upward from ground level. Its height above the ground (in meters) after t seconds is modeled by the polynomial
s(t)=−2t4+16t3−24t2+10,
for 0≤t≤5. Use this model to analyze the rocket’s motion.
(a) Find the velocity v(t) of the rocket.
(b) Determine all times in the interval 0≤t≤5 at which the rocket’s velocity is zero, and interpret each of these times in the context of the flight.
(c) For each time found in part (b), decide whether the rocket is at a local maximum height, a local minimum height, or neither. Support your answer with an analysis of the sign of v(t) on appropriate intervals.
(d) Identify the absolute maximum height attained by the rocket on the interval 0≤t≤5 and the time at which it occurs. Justify your answer, including consideration of the interval endpoints.