The position function s(t)=t3−8ts(t)=t3−8t gives the position in miles of a freight train where east is the positive direction and tt is measured in hours. determine the direction the train is traveling when s(t)=0.s(t)=0. determine the direction the train is traveling when a(t)=0.a(t)=0. determine the time intervals when the train is slowing down or speeding up.

we are given

(a)

we can set s(t)=0

and then we can solve for t

so, we get

............Answer

(b)

Firstly, we will find a(t)

we know that a(t) is second derivative of s(t)

now, we can find derivative again

now, we can set it to 0

and then we can solve for t

............Answer

(c)

Firstly, we will set s'(t)=0 and a(t)=0

and then we solve for t

now, we can draw number line

and then we can check sign of s'(t) and s''(t) on each intervals

same sign means speeding up

different sign means speeding down

speeding up interval:

speeding down interval:

answerthe vertex form

step-by-step explanation: