Suppose that at time ` t ul( gt)0` the position of a particle moving on the x-axis is `x=(t-1)(t-4)^(4)m`.
(a) When is the particle at rest ?
(b) During what time interval does the particle move to the left ?
(c) Find maximum velocity of particle while moving to the left.

Explanation

`(a) (dx)/(dt)=(t-4)^(4)+4(t-4)^(3)(t-1)=0`
`:. " " (t-4)^(3)[t-4+4t-4]=0`
`rArr " " t=4s or t=(8)/(5)s`
(b) `(dx)/(dt) lt 0 rArr (t-4)^(3)(5t-8)lt0`
For this , `tlt 4`
and `t gt (8)/(5)`
`rArr " " t in ((8)/(5),4)`
(c) Particle will be faster when `((dx)/(dt))` is maximum
`:. " " (d)/(dt)((dx)/(dt))=0`
`:. " " (d^(2)x)/(dt^(2))=3(t-4)^(2)(5t-8)+5(t-4)^(3)=0`
`rArr " " (1-4)^(2)[15t-24+5t-20]=0`
`rArr " " t=4s " " or" " t=(44)/(20)=2.2s`
But at `t=4s(dx)/(dt)=0`
`:. ((dx)/(dt))` is maximum at t=2.2 s
and maximum value of `(dx)/(dt)=(2.2-4)^(3)(5xx2.2-8)`
`=-17.49m//s`