$1.$ Given a cubic spline interpolation:

S$($x$)=\{[$ S$\_0($x$)=1-2$ x$-$x$^30<=$ x$<1$; S$\_1($x$)=2+$b$($x$-1)+$c$($x$-1)^2+$d$($x$-1)^21<=$ x$<2].$

determine constants b$,$ c$,$ and d so that all conditions for a natural cubic spline hold.