(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