A high-tech company purchases a new computing system whose initial value is V. The system will depreciate at the rate f = f(t) and will accumulate maintenance costs at the rate g = g(t), where t is the time measured in months. The company wants to determine the optimal time to replace the system.

(a) Let

C ( t ) = 1 t ∫ 0 t [ f ( s ) + g ( s ) ] d s

Show that the critical numbers of C occur at the numbers t where C(t) = f(t) + g(t).

(b) Suppose that

f ( t ) { V 15 − V 450 t    if 0 <  t   ⩽ 30   0                  if  t > 30    

and

g ( t ) = V t 2 12 , 900 t > 0

Determine the length of time T for the total depreciation D ( t ) = ∫ 0 t f ( s )   d s to equal the initial value V.

(c) Determine the absolute minimum of C on (0, T].

(d) Sketch the graphs of C and f + g in the same coordinate system, and verify the result in part (a) in this case.