improvement in rather evident. As additional check we have considered the equality $$1 - \int_{0}^{1} dx\, q(x) = T$$ which is satisfied by our numerical solution for all studied temperatures with at least four digits. For example we for $T=0.8$ we get $0.79999(4)$, while for $T=0.5$ the value is $0.49999(3)$.

Note that not only by fine tuning of $x_0$ we can have a good solution for $q(x)$ at the given temperature, but we also have [*the value*]{} of $x_0$. This is best seen by analyzing the concavity of $q(x)$ near $x_0$. In Figure \[fig:q2x.T0.4-x0-500\] we show the second derivative of $q(x)$ near $x_0$ for $T=0.4$ and $N_x=500$, from which one may conclude that $0.505 < x_{\rm max} < 0.510$.

A careful analysis of the stability of this results as function of $N_x$, see Figure \[fig:q2x.T0.4-x0.510-0.515\], reveals, however, that the correct estimation is $0.510 < x_{\rm max} < 0.515$, in rather good agreement with the analytical result $x_{\rm max} = 0.5111\pm 0.0002$. The same analysis for $T=0.6$ leads to $0.438 < x_{\rm max} < 0.440$.

We are now in the position of checking the results of previous section about the derivative of the order parameter at $x=0$, and in particular the 