Maxima, minima and optimisation sheet

Essay frame

A firm has R(x) = 150x − 0.2x² and C(x) = 2,000 + 40x + 0.05x². Find the output that maximises profit, prove it is a maximum, and find the maximum profit and the price.

How to lay out the answer. Follow the frame, then write it in your own words.

1 · Form the profit function. P(x) = 110x − 0.25x² − 2,000.
2 · Differentiate and set to zero. P′(x) = 110 − 0.5x = 0, so x = 220.
3 · Second-derivative test. P″(x) = −0.5 < 0: a maximum.
4 · Maximum profit. P(220) = 24,200 − 12,100 − 2,000 = Rs. 10,100.
5 · Price. p = 150 − 0.2(220) = Rs. 106.
6 · Check MR = MC. MR = 150 − 0.4(220) = 62; MC = 40 + 0.1(220) = 62. ✓
7 · Sketch. [Profit curve: hill-shaped, peak at (220, 10,100)]
8 · Business link. Producing beyond 220 units adds more cost than revenue.