Stationary Points and Optimisation NCEA Practice Questions | Calc.nz

Skill overview

What these questions practise

The matched Level 2 and Level 3 methods include derivative sign, first- and second-derivative classification, graph interpretation, and optimisation in geometric or contextual domains.

Relevant standards: NCEA Level 2 Calculus — AS91262 · NCEA Level 3 Differentiation — AS91578

This collection contains 43 matching walkthroughs across 10 examination years.

The grouping is based on the method recorded for each walkthrough. Use the linked NZQA standard and paper material for authoritative wording, diagrams, and assessment information.

Before you start

Common mistakes in these methods

Solving the derivative condition finds a candidate; use the original function when a coordinate is also required.
Do not label a maximum, minimum, or stationary inflection without the sign or derivative evidence requested by the question.
Check the model's domain and interpret the admissible optimum in context rather than reporting every algebraic candidate.

Related practice

Stationary Points and Optimisation questions from different years

Open a question for hints and a progressive worked solution, or use the standard and paper links to return to its assessment context.

NCEA Level 2 · AS91262 2025 Calculus 6 questions

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Question 1(c) — Stationary Points and Optimisation Finding critical points, identifying the minimum, and solving for the constant. Question 2(a) — Stationary Points and Optimisation Sketching the derivative from stationary points, signs, and changes in concavity. Question 2(c) — Stationary Points and Optimisation Differentiating, evaluating a rate, and using derivatives to justify the minimum turning point. Question 2(d) — Stationary Points and Optimisation Using a stationary point and an x-intercept to find constants, then classifying the other stationary point. Question 3(c) — Stationary Points and Optimisation Using a surface-area constraint, writing volume in one variable, maximizing it, and proving the maximum. Question 3(d) — Stationary Points and Optimisation Differentiating, factorising, classifying critical points, and finding where the function is decreasing.

NCEA Level 3 · AS91578 2025 Differentiation 4 questions

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Question 1(e) — Parametric Differentiation Parametric differentiation with stationary inflection points. Question 3(a) — Stationary Points and Optimisation Reading differentiability, stationary points, and a limit from a graph. Question 3(b) — Stationary Points and Optimisation Stationary points on a logarithmic curve. Question 3(e) — Stationary Points and Optimisation Maximising the area of a rectangle inside a curve.

NCEA Level 3 · AS91578 2024 Differentiation 4 questions

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Question 1(d) — Stationary Points and Optimisation Stationary points, the second derivative test, and classifying the point. Question 2(e) — Stationary Points and Optimisation Using a discriminant condition to force a single turning point. Question 3(c) — Quotient Rule Quotient rule differentiation and finding stationary points. Question 3(e) — Stationary Points and Optimisation Forming an area function and proving its largest possible value.

NCEA Level 3 · AS91578 2023 Differentiation 2 questions

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Question 1(e) — Stationary Points and Optimisation Maximising a triangle area on the curve y = x(x - 2m) squared. Question 3(d) — Stationary Points and Optimisation Stationary points and classifying them with the second derivative.

NCEA Level 3 · AS91578 2022 Differentiation 5 questions

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Question 1(b) — Stationary Points and Optimisation Finding and interpreting stationary points. Question 2(c) — Quotient Rule Quotient rule and finding when motion is stationary. Question 2(d) — Stationary Points and Optimisation Forming an area function and maximising it. Question 3(d) — Stationary Points and Optimisation Finding stationary points and classifying them. Question 3(e) — Stationary Points and Optimisation Building a time function and minimising it.

NCEA Level 3 · AS91578 2021 Differentiation 6 questions

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Question 1(b) — Stationary Points and Optimisation Reading stationary points, concavity, and limits from a graph. Question 1(c) — Chain Rule Product and chain rules for stationary points. Question 1(e) — Stationary Points and Optimisation Cone and cylinder optimisation with a maximum-volume proof. Question 2(b) — Quotient Rule Quotient rule and stationary points. Question 3(b) — Stationary Points and Optimisation Stationary point on a radical function. Question 3(e) — Chain Rule Lamp and table optimisation using product and chain rules.

NCEA Level 3 · AS91578 2020 Differentiation 4 questions

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Question 1(c) — Quotient Rule Quotient rule and a stationary point on a logarithmic function. Question 1(e) — Stationary Points and Optimisation Maximising a cylinder volume inside a sphere. Question 2(c) — Chain Rule Product and chain rules for stationary points. Question 3(d) — Stationary Points and Optimisation Stationary points and second derivative classification.

NCEA Level 3 · AS91578 2019 Differentiation 1 question

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Question 3(c) — Stationary Points and Optimisation Maximising a rectangle area under y = 4 - square root of x.

NCEA Level 3 · AS91578 2018 Differentiation 3 questions

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Question 2(d) — Product Rule Product rule and stationary points of an exponential function. Question 3(c) — Stationary Points and Optimisation Maximising the area of a triangle inside a parabola. Question 3(e) — Stationary Points and Optimisation Building and minimising a symmetric wire-length model.

NCEA Level 3 · AS91578 2017 Differentiation 4 questions

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Question 1(e) — Derivative Conditions Combining point and turning-point conditions to find parameters. Question 2(b) — Stationary Points and Optimisation Maximising a logarithmic germination model over its domain. Question 2(d) — Stationary Points and Optimisation Minimising squared distance from a point to a curve. Question 2(e) — Stationary Points and Optimisation Maximising the area of a rectangle in a semicircle.

NCEA Level 3 · AS91578 2016 Differentiation 4 questions

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Question 2(c) — Stationary Points and Optimisation Continuity, differentiability, stationary points, concavity, and limits from a graph. Question 2(e) — Stationary Points and Optimisation Maximising the volume of a cone inside a sphere. Question 3(c) — Stationary Points and Optimisation Maximising a rectangle area beneath a parabola. Question 3(e) — Stationary Points and Optimisation Maximising a rugby conversion angle using trigonometry and calculus.

Keep practising

Chain rule Product and quotient rules Related rates Parametric differentiation

Page updated 9 August 2026.