C1 = 200; 
C2 = 100; 
LTCR[
          T_] := (((C1 - 10 (2 - T)) (2 - T) + (C1 + C2) (T - 1))/(-0.5 (T^2) + 2  T - 0.5));

Solve[D[LTCR[T], {T, 1}] == 0, T]

LTCR[1.37123]

LTCR[2]