(a)

  from the given problem we can see that at point A the weight of the coaster must be just large
   enough to supply the centripetal acceleration
   so we get
   m (v_A² / R) = m g
   v_A² = R g
   so applying the conservation of mechanical energy from the start point A we get
   (1 / 2) m v_o² + m g h = (1 / 2) m v_A² + m g (2 h / 3)
   v_o² = v_A² - (2 g h / 3)
   so by using v_A² = R g
   the car barely stays on the track at A gives
   v_o = √[R g - (2 g h / 3)]
(b)
   if the speed of the coaster is to be zero at point B conservation of mechanical energy from the
   start to point B gives
   0 + m g h' = (1 / 2) m v_o² + m g h
                   = (1 / 2) m {g [R - (2 h / 3)]} + m g h
   h' = (R / 2) + (2 h / 3)