Define the quantity to maximise or minimise, with units.

Use the constraint to express it in one variable and state the feasible domain.

Find all stationary candidates inside that domain.

Compare the objective values at relevant candidates and included endpoints. If an endpoint is excluded, examine the limiting behaviour.

Justify the result and answer the question in context.

For \(A(x)=12x-x^2\), \(0\le x\le4\), the stationary value \(x=6\) is outside the allowed interval. Since \(A'(x)=12-2x>0\) throughout, the maximum is \(A(4)=32\).

The second-derivative test proves a local extremum. A global maximum or minimum over a restricted domain still requires comparison with all relevant candidates and boundary values. For lengths, volumes and angles, enforce the physical restrictions.

Problems Involving Maximisation and Minimisation

Differentiation

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Optimisation and Endpoints

Optimisation and Endpoints