I've started studying general relativity course and now I have a question about proper time. Consider functional $$S[x]=-\int_A^B ds,$$ where $A$, $B$ are fixed points of the space-time and $ds^2=dt^2-dx^2$ (let our space-time be two-dimensional without loss of generality). Finding its minimum is equivalent to maximizing proper time $s=\int_A^B ds$ and it a well-known fact that maximizers of proper time are straight lines, as it can be easily checked by writing Lagrange equation for $\mathcal{L}(t,x,\dot{x})=-\sqrt{1-\dot{x}^2}$. So my question is: what are minimizers of proper time?

I've heard that there are, hm, lots of them, but I can't write down any. If one wants to minimize $s$, he takes lagrangian $\mathcal{L}=\sqrt{1-\dot{x}^2}$ and he obtains that $$\ddot{x}=0,$$ so minimizers necessarily are straight lines. But we have already proved that they are maximizers, so I come to contradiction. Where am I wrong?