  We consider a double power nonlinear Schr\"odinger equation which possesses
the algebraically decaying stationary solution $\phi_0$ as well as
exponentially decaying standing waves $e^{i\omega t}\phi_\omega(x)$ with
$\omega>0$. It is well-known from the general theory that stability properties
of standing waves are determined by the derivative of $\omega\mapsto
M(\omega):=\frac{1}{2}\|\phi_\omega\|_{L^2}^2$; namely $e^{i\omega
t}\phi_\omega$ with $\omega>0$ is stable if $M'(\omega)>0$ and unstable if
$M'(\omega)<0$. However, the stability/instability of stationary solutions is
outside the general theory from the viewpoint of spectral properties of
linearized operators. In this paper we prove the instability of the stationary
solution $\phi_0$ in one dimension under the condition
$M'(0):=\lim_{\omega\downarrow 0}M'(\omega)\in[-\infty, 0)$. The key in the
proof is the construction of the one-sided derivative of
$\omega\mapsto\phi_\omega$ at $\omega=0$, which is effectively used to
construct the unstable direction of $\phi_0$.
