The position of a particle moving along the x-axis is given by #x(t)=t^3-9t^2+24t# meters where t is in seconds, #t>=0# a) Draw a sign diagram for the particle's velocity and acceleration functions.?

The velocity is the derivative of the position

#x(t)=t^3-9t^2+24t#

#v(t)=3t^2-18t+24=3(t^2-6t+8)#

#=3(t-2)(t-4)#

The acceleration is the derivative of the velocity

#v(t)=3t^2-18t+24#

#a(t)=6t-18=6(t-3)#

The sign chart for the velocity is as follows :

#color(white)(aaaa)##t##color(white)(aaaaa)##0##color(white)(aaaaaaaa)##2##color(white)(aaaaaaaa)##4##color(white)(aaaaaa)##+oo#

#color(white)(aaaa)##t-2##color(white)(aaaaaa)##-##color(white)(aa)##0##color(white)(aaa)##+##color(white)(aaaaaa)##+#

#color(white)(aaaa)##t-4##color(white)(aaaaaa)##-##color(white)(aa)####color(white)(aaaa)##-##color(white)(aaa)##0##color(white)(aa)##+#

#color(white)(aaaa)##v(t)##color(white)(aaaaaa)##+##color(white)(aaa)##0##color(white)(aaa)##-##color(white)(aaa)##0##color(white)(aa)##+#

#color(white)(aaaa)##s(t)##color(white)(aaaaaa)##↗##color(white)(aa)##20##color(white)(aaa)##↘##color(white)(aa)##16##color(white)(aa)##↗#

The sign chart fot the acceleration is as follows :

#color(white)(aaaa)##"Interval"##color(white)(aaaa)##(0,3)##color(white)(aaaa)##(3,+oo)#

#color(white)(aaaa)##a(t)##color(white)(aaaaaaaaa)##-##color(white)(aaaaaaa)##+#

#color(white)(aaaa)##s(t)##color(white)(aaaaaaaaa)##nn##color(white)(aaaaaaa)##uu#

graph{(y-(x^3-9x^2+24x))(y-(3x^2-18x+24))(y-(6x-18))=0 [-23.9, 41.07, -6.7, 25.78]}