Applications of the derivative
The applications block turns derivatives into answers about shape and about economic decisions: intervals of increase and decrease, maxima and minima, concavity, inflection points, and the optimisation of cost, revenue and profit functions.
Core rules
- f is increasing where f' > 0 and decreasing where f' < 0; solve f' = 0 for the critical points and read the sign of f' on each subinterval.
- At a critical point, a change from + to − gives a maximum, from − to + a minimum, and no change an inflection with horizontal tangent. The second derivative test works when f'' ≠ 0: f'' < 0 maximum, f'' > 0 minimum.
- f'' > 0 means convex (∪), f'' < 0 concave (∩). An inflection point needs f'' = 0 (or undefined) and a change of sign of f''.
- On a closed interval, compare the values at the critical points and at both endpoints.
- Economics: a marginal magnitude is a derivative. Profit is maximised where marginal revenue equals marginal cost with the profit condition satisfied; average cost AC = C(q)/q is minimised where AC' = 0, which always occurs at the point where AC = MC.
Worked example
A firm has C(q) = q² + 4q + 100 euros, q > 0. Average cost is AC(q) = q + 4 + 100/q. Then AC'(q) = 1 − 100/q², which vanishes at q = 10 (the negative root is rejected). Since AC''(q) = 200/q³ > 0, q = 10 is a minimum with AC(10) = 24 euros per unit. Check with marginal cost: MC = C'(q) = 2q + 4, so MC(10) = 24 = AC(10), the standard consistency check the exam expects you to quote.
Traps that cost marks
- Concluding an extremum from f' = 0 alone; x³ at x = 0 has neither.
- Reading f'' > 0 as concave (it is convex).
- Calling a point an inflection without verifying the sign change of f''.
- Ignoring the economic domain q ≥ 0 or the endpoints of a closed interval.
- Maximising revenue when the question asked for profit, or forgetting fixed costs in the profit function.