A factory produces custom bicycle frames. Daily revenue in dollars from selling x frames is modeled by R(x)=450x. Daily cost in dollars consists of three components:
• Base cost: $18,000.
• Labor adjustment: Clab(x)=75∣x−200∣.
• Materials: 110x.
Thus the total daily cost is C(x)=18000+75∣x−200∣+110x for 0≤x≤400.
(a) Write the profit function P(x) and express it without absolute value by giving an explicit piecewise definition valid on [0,400].
(b) Calculate all critical numbers of P on [0,400], showing derivative computations for each piece of the domain.
(c) Determine the production level that maximizes daily profit and compute the corresponding maximum profit.
(d) Interpret your result in practical terms for the factory’s production planning.