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 is the time measured in months. The company wants to determine the optimal time to replace the system.
(a) Let Show that the critical numbers of C occur at the numbers t where C (t) = f (t) + g (t).
(b) Suppose that and

Determine the length of time for the total depreciation D (t) = f(s) to equal the initial value V.
(c) Determine the absolute minimum of 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.

C) -[/) + g(0]ds t if 0 30 Vr? g(1) t>0 12,900