ClearAll;
R[t_]:= Which[t<1,(t-t^3)/4,t>=1,(1-t)/(1+t)];
RPrime[t_]:= Which[t<1,(1-3t^2)/4,t>=1,-2/(1+t)^2];
Show[GraphicsArray[{{Plot[R[t],{t,0,3}]},{Plot[RPrime[t],{t,0,3}]}}]]

(* Find the global maximum of R *)
FindMinimum[-R[t],{t,0,3}]