Problem 78E

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)=1t0t[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)={V15V450tif0t300ift30 and g(t)=Vt212,900t0 Determine the length of time T for the total depreciation D(t)=0tf(s)ds 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.

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