A shipping company charges for express delivery according to the total mass m (in kilograms) of the package. The cost function C (in dollars) for packages that weigh between 0 kg and 15 kg is given by
C(m)=⎩⎨⎧8m+12,44+5(m−4),69+3(m−10),0≤m≤4,4<m≤10,10<m≤15.
Answer the following questions related to the company’s pricing policy.
a. Part 1: (a) State the open intervals on which C is differentiable and justify your answer using the definition of differentiability.
b. Part 2: (b) Explain why the absolute minimum cost for a package in the domain 0≤m≤15 must occur at either an endpoint of the domain or at m=4 or m=10. No numerical calculation is required.
c. Part 3: (c) Describe, in practical terms, what information a customer would need to determine the least expensive mass for repackaging a shipment, based on your reasoning in part (b).