Apoint moves on the x-axis in such a way that its velocity at time t (t> 0) is given by v=lnt/t. at which value of t does v attain its maximum?
(a) 1
(b) e^(1/2)
(c) e
(d) e^(3/2)
(e) there is no maximum value for v.

(D) e^(3/2)

v = ln(t) / t 

max of v when dv/dt = 0 
dv/dt = (t/t - ln(t))/t² 
= [1 - ln(t)]/t² = 0 
ln(t) = 1 
t = e (option C) 

check this is a max and not a min 
d²v/dt² = [t²(-1/t) - 2t(1 - ln(t))]/t^4 
= [-1 - 2t + 2t ln(t)]/t^4 

= [-1 - 2e + 2e ln(e)]/e^4 
= [-1 - 2e + 2e]/e^4 
= -1/e^4 < 0 so it is a max