Calculus · step 4 of 13

Extrema & optimisation

Interior extrema occur at critical points, where f′ = 0 or f′ is undefined — but not every critical point is an extremum, as x³ at 0 shows. On a closed interval you must also check the endpoints, since the extreme value theorem guarantees a maximum and minimum but says nothing about where. Applied problems add one step: build the function and eliminate variables using the constraint.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Set up the objective and constraintFor word problems, write what is to be optimised and the relation between the variables, then reduce to one variable.
Find the critical pointsSolve f′(x) = 0 and note any point where f′ fails to exist within the domain.
Classify themSecond derivative test: f″ > 0 means a local minimum, f″ < 0 a local maximum, f″ = 0 is inconclusive. Or use a sign chart for f′.
Check the endpoints and state the domainOn a closed interval compare the critical values with the endpoint values. The largest wins.

Worked example

A rectangle has perimeter 20. Maximise its area.

Let the sides be x and y with 2x + 2y = 20, so y = 10 − x and 0 < x < 10.
Area A(x) = x(10 − x) = 10x − x².
A′(x) = 10 − 2x = 0 gives x = 5.
A″(x) = −2 < 0, so this is a maximum. Endpoints give A → 0.

Answer. The maximum area is 25, attained by the 5 × 5 square.

Where marks get dropped

These are the specific errors that cost credit on extrema & optimisation questions — QED's rubric penalises each of them separately.

Assuming every critical point is an extremum. f(x) = x³ has f′(0) = 0 but a point of inflection, not a turning point.
Forgetting the endpoints on a closed interval, where the true maximum frequently sits.
Failing to state the domain in a word problem. Negative lengths are not admissible, and the domain determines whether endpoints matter.

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Extrema & optimisation — frequently asked questions

What if the second derivative is zero?

The test is inconclusive — fall back on a sign chart for f′ around the point, or examine higher derivatives.

Does the extreme value theorem guarantee a maximum?

Yes, for a continuous function on a closed bounded interval. On an open interval or with a discontinuity, no maximum need exist.

How do I know I have a global maximum?

On a closed interval compare all critical and endpoint values. On an open domain, argue about behaviour at the boundaries or use concavity throughout.

The rest of Calculus

Limits, derivatives, integrals and their applications. Each subtopic below has its own method, worked example and mark-losing traps.

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