Help with derivative application and application in relation to time.
I have a Calculus test tomorrow, and I'm honestly lost here. This is from a sample test.
5. The function s(t)= -2t^3 + 21t^2 - 60t + 52 gives the distance (inches) of an object from a certain fixed point as a function of time (seconds). Assume the object is moving in one dimension (which means a straight line path…i.e. this is rectilinear motion).
a) Does the object ever reach the “certain fixed point” (if so tell when)? Answer yes or no and describe how you got your answer. (6 points)
I have no idea how to figure this part out.
b) Let’s say the object is advancing when it’s moving towards the fixed point and retreating when it’s moving away from the fixed point. Using these definitions, determine time intervals when the object is advancing and retreating. (8 points)
Derivative is used here, if I'm not mistaken.
s`(t)= -6t^2 + 42t - 60
No idea where to go from there.
c) Let’s say acceleration happens when speed increases and deceleration happens when speed decreases. Using these definitions, determine time intervals when the object is accelerating and decelerating.
Second derivative here, I'm pretty sure. No idea what to do with it.
s``(t)= -12t + 42
I need some serious help here.