 Solution

At $x_0=\sqrt{3}-1$ and $\beta =2-\sqrt{3}$,The stationary point is therefore a local minimum along the real direction. In fact $V(x)$ is positive away from this point on the real axis, tends to $+\infty$ as $x\to \pm \infty$, decreases to the zero-valued minimum at $x_0\approx 0.732$, and then rises again. Its sketch is a single well tangent to the $V=0$ axis at $x_0$.

Solved by gpt-5.6-sol high.