A device is currently new. We replace it with an identical new device either when it breaks or at the fixed time \(T\), whichever comes first. The lifetime of a device has the Weibull density:

The cost incurred due to breakdown is twice as much as the cost due to a simple replacement of a functioning device. Show that to minimize long-run average cost per unit time, it is optimal never to replace the device. (Note that the inter-renewal time has a density on \((0, T)\) and puts positive mass on the point \(T\). Use a sensible extension of the definition of expectation for this mixed discrete-continuous random variable.)

f(x)= {7x120 x-1/2-x if x>0 otherwise