A company produces and sells a single product. The price‐demand relationship is modeled by p(x)=120−0.5x, where x is the number of units sold and p is the price per unit in dollars. The total cost of producing x units is C(x)=30x+400, valid for 0≤x≤200.
(a) Write the revenue function R(x) and the profit function P(x).
(b) Find all critical points of the profit function on the closed interval [0,200].
(c) Determine the production level that maximizes profit, verify that it is indeed a maximum, and state the maximum profit and the corresponding price per unit.
(d) If the company were to increase the selling price by exactly \$5 per unit while keeping production at the profit‐maximizing level found in part (c), would profit increase or decrease? Justify your answer with appropriate calculations or reasoning.