Application of Derivatives: Maxima, Minima and Word Problems

The translation step, done properly

Every optimisation word problem has the same skeleton, and teaching it as a skeleton removes most of the fear.

• Name what is being maximised or minimised and write it as a formula, even with two variables in it.
• Find the constraint — the sentence in the question that relates the two variables.
• Use the constraint to eliminate one variable. Now there is a function of one variable, and the calculus can start.
• Note the domain. A length cannot be negative, and questions are set where the unconstrained answer falls outside the sensible range.

First derivative or second?

Either is acceptable, and students should pick one and use it consistently. The first derivative test with a sign table is more reliable when the second derivative is messy; the second derivative test is faster when it is clean. Switching between them mid-paper is where confusion enters.

Whichever is used, the conclusion has to be stated in words — that the point is a maximum, and what its value represents in the original problem. Answers that end at a number without returning to the question lose the final mark.

The other applications

Rate of change, increasing and decreasing intervals, and tangents and normals are the rest of the chapter and are more mechanical. The common losses there are sign errors in the sign table and forgetting that a tangent question usually wants the equation of a line, not merely the gradient.

How to practise

Word problems specifically, and in volume. A student who has done fifteen optimisation word problems recognises the skeleton instantly; a student who has done three is translating from scratch every time under exam pressure.

Questions parents ask