Mathematics

Problem

Let

k

be a positive constant. The position of a particle moving along a line is given by

s(t)=t^{3}-3k t^{2}+2k^{2} t, \quad t\ge 0,

where

s

is measured in meters and

t

in seconds. (a) Express the velocity

v(t)

in terms of

k

and

t

. (b) In terms of

k

, determine all critical times

t>0

at which the particle could achieve an extreme position. (c) For

k=4

, decide whether each critical time from part (b) corresponds to a local maximum, a local minimum, or neither. Provide justification. (d) For general

k>0

, find an explicit formula for the particle’s maximum position on the interval

[0,\dfrac{3k}{2}]