The position is given by `s(t)=10+2t^7-t^9` for t>0.
The velocity function is the first derivative of the position function. The acceleration is the derivative of the velocity function (or the second derivative of the position function.) The velocity function takes on its maximum or minimum when the acceleration function is zero.
(a) The velocity is zero when `14t^6-9t^8=0`
`t^6(14-9t^2)=0==> t^6= 0,(14-9t^2)=0`
since t>0 we consider `14-9t^2=0 ==> t^2=14/9`
Again since t>0 we have `t=sqrt(14)/3~~1.25`
The velocity is zero when t is approximately 1.25
(b) The acceleration is zero when `84t^5-72t^7=0`
`12t^5(7-6t^2)=0 ==> 12t^5=0,7-6t^2=0`
Since t>0 we have `t^2=7/6==>t=sqrt(7/6)~~1.08`
The acceleration is zero at t approximately 1.08
(c) The velocity is maximum when the acceleration is zero. For positive t, this occurs when t is approximately 1.08 and the velocity is approximately 5.56
The graph of the position function in black, the velocity function in red, and the acceleration in green: