Question: Wickham's Wicker Widgets produces a micro-mini-widget. The revenue function associated with manufacturing and distributing this product is as follows: \(2{{x}^{2}}+390x-3456\), where x is the number of batches of micro-mini-widgets sold; the cost function associated with manufacturing and distributing this product is as follows: \(3{{x}^{2}}+210x+1728\).

(a) Construct the profit function \(P\left( x \right)\), simplifying as necessary.

(b) Determine the profit or loss associated with manufacturing and distributing 20 batches of these widgets.

(c) Determine the profit or loss associated with manufacturing and distributing 80 batches of these widgets.

(d) Determine the profit or loss associated with manufacturing and distributing 140 batches of these widgets.

(e) Determine the profit or loss associated with manufacturing and distributing 200 batches of these widgets.

(f) Determine the break-even points for the manufacturing and distributing of these widgets.