A point 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

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A point 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.

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A point 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.