Aparticle moves along a straight line and its position at time t is given by
s(t)=2t^3−24t^2+90t t> =0 where s is measured in feet and t in seconds.
(b) use interval notation to indicate the time interval(s) when the particle is speeding up and slowing down.
(a) use interval notation to indicate the time interval or union of time intervals when the particle is moving forward and backward

(A) Use interval notation to indicate the time interval or union of time intervals when the particle is moving forward and backward

the particle is speeding up

if  a(t) >0, it means 12t-48>0, and t >4,  I = ]4, infinity[

the particle is speeding down

if  a(t) < 0, it means 12t-48>0, and t < 4,  I = ] -infinity, 4[

when the particle is moving forward and backward

that is depending of the sign of v(t)

moving forward if v(t)> 0

moving backward if v(t)<  0

but v(t) = 6t² - 48 t +90, this quadratic equation has exactly two solutions, t =1, and t =3

so  after taking care the sign of 6t² - 48 t +90,

v(t)> o when  t is in I = ] –inf, 1[ U ]4, inf[,the particle is moving forward

v(t) < o when  t is in I = ]1, 4[,the particle is moving backward

grav potential energy