We study the stability and exact multiplicity of periodic solutions of the Duffing equation with cubic nonlinearities,
(
∗
)
x
+
c
x
′
+
a
x
−
x
3
=
h
(
t
)
,
\begin{equation*} x+cx’+ax-x^{3}=h(t),\tag {$*$} \end{equation*}
where
a
a
and
c
>
0
c>0
are positive constants and
h
(
t
)
h(t)
is a positive
T
T
-periodic function. We obtain sharp bounds for
h
h
such that
(
∗
)
(*)
has exactly three ordered
T
T
-periodic solutions. Moreover, when
h
h
is within these bounds, one of the three solutions is negative, while the other two are positive. The middle solution is asymptotically stable, and the remaining two are unstable.