A small workshop produces and sells a handcrafted item
MATHEMATICS · Subjective · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A small workshop produces and sells a handcrafted item. The demand analysis suggests that the price (in rupees) at which x units can be sold is p=100−10x2, while the cost (in rupees) of producing x units is given by C(x)=20x+40.
Let the revenue be R(x)=px and the profit be P(x)=R(x)−C(x).
(a) Express R(x) and P(x) in terms of x only.
(b) Find all critical points of the profit function P.
(c) Classify each critical point identified in part (b), stating clearly whether it gives a local maximum profit, a local minimum profit, or neither.
(d) Determine the maximum possible profit and the corresponding number of units that should be produced and sold to achieve it. Round your answer for x to the nearest whole unit if necessary.