= Solution
The direct <First Lyapunov theorem> says that a continuously differentiable positive-definite function $V$ near an equilibrium, with $\dot V\leq0$, proves stability. <LaSalle invariance principle> says that in a <compact> positively invariant set, trajectories approach the largest invariant subset of $\{\dot V=0\}$. Under the terminology in which “first theorem” means the linearization theorem, <eigenvalues> with negative real part imply <asymptotic stability>, while zero-real-part <eigenvalues> leave the test inconclusive. Here the linearization has <eigenvalues> $0,-k$, so the direct criterion is needed.
Write $y=\dot x$ and choose
$$
V(x,y)=\frac{y^2}2+\int_0^x\sin^3s\,ds
=\frac{y^2}2+\frac23-\cos x+\frac13\cos^3x.
$$
Near zero, $V=y^2/2+x^4/4+O(x^6)$ is positive definite. Along solutions,
$$
\dot V=y(-ky-\sin^3x)+\sin^3x\,y=\boxed{-ky^2\leq0}.
$$
The component containing the origin of a sufficiently small closed sublevel set is <compact>, lies in $|x|<\pi$ and is positively invariant. The set $\dot V=0$ has $y=0$; to remain there also requires $\sin^3x=0$, so its largest invariant subset is just the origin. The two theorems establish \b[the origin is asymptotically stable].
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