I think the answer may be (d). Here is why:
is unbounded, then
. the integral blows up as f'' does, so
is also unbounded.
Since there is a unique c s.t.
, then we know
which is true because of the mean value theorem. Then, f'(c) < 0, f'(x) is strictly increasing for x >=c, and f'(x) is unbounded. Therefore, f'(x) has a unique zero.
Therefore, only II,III hold for
. Then, using the exact same arguments as above we can show that only II,III hold for