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# Optimal time to replace

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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 the integral from 0 to t of [f(s)+g(s)]ds. 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 is less than t and t is less than or equal to 30; f(t)=0 if t is greater than 30; and g(t)=(Vt^2)/12900 if t is greater than 0. Determine the length of time T for the total depreciation D(t)=the integral from 0 to t of f(s)ds to equal the initial value V.

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