Essential Steps: Problems on Maximum and Minimum Values

Use the constraint to express the quantity to optimise in one variable. State the feasible domain.

Find the interior stationary points and test their nature. Compare admissible endpoints and relevant limits for a global maximum or minimum.

Discard infeasible values. An excluded endpoint may give a limiting value that is never attained.

Mensuration formulae

Substitute back to find the requested quantity and include units. Add bases when total surface area is required.

Example

A rectangle has perimeter \(20\) cm and width \(x\) cm, so its area is \(A=x(10-x)\) with \(0<x<10\).

\(\dfrac{\mathrm dA}{\mathrm dx}=10-2x=0\Rightarrow x=5;\qquad \dfrac{\mathrm d^2A}{\mathrm dx^2}=-2<0\), so the greatest area is \(5\times5=25\text{ cm}^2\).