A Level · Cambridge (CIE) · 9709

Locate stationary points and determine their nature

Solve the zero-gradient equation, evaluate the original curve and use a valid second-derivative or sign test to justify a sketch or an optimum.

Pure Mathematics 1 · Differentiation · 1.7

Pure Mathematics 1 (Paper 1) · AS & A Level Mathematics 9709, 2026–2027. Use an informal understanding of limits; formal general differentiation from first principles and points of inflexion are not required in section 1.7. Written practice is self-checked.

Before you begin

Differentiate polynomials twice and solve factored equations.
Interpret derivative signs and the difference between a local feature and a whole-domain maximum.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

Which equation locates a stationary input on a differentiable curve?

What you will learn

Find stationary point coordinates and identify local maxima or minima.
Use the second derivative or a derivative sign change, and compare endpoints for a constrained optimum.

For y = x³ − 6x² + 9x + 1, the stationary points are a local maximum (1, 5) and a local minimum (3, 1), with y-intercept (0, 1). The curve rises for x < 1, falls for 1 < x < 3 and rises for x > 3. Its second derivatives at the stationary inputs are −6 and 6 respectively. The plotted window is −0.25 ≤ x ≤ 4.25; the curve continues beyond it, so the local maximum is not a greatest value over the entire real domain.

First locate the point; then justify its nature

A stationary point on y = f(x) has finite first derivative f′(x) = 0. Solve that equation for the input and substitute into the original function for the height. If f″(a) > 0, a stationary point is a local minimum; if f″(a) < 0, it is a local maximum. A positive-to-negative first-derivative sign change also gives a maximum, and negative-to-positive gives a minimum.

For f(x) = x³ − 6x² + 9x + 1, f′(x) = 3(x − 1)(x − 3), so the stationary inputs are 1 and 3. Their points are (1, 5) and (3, 1). Since f″(x) = 6x − 12, its values −6 and 6 show a local maximum at (1, 5) and a local minimum at (3, 1). Together with the y-intercept (0, 1), the derivative signs and the positive cubic leading term, these facts support a rising–falling–rising sketch.

A zero second derivative is inconclusive

For y = (x − 2)⁴ + 1, the first derivative is 4(x − 2)³ and the second is 12(x − 2)². Both vanish at x = 2, so the second-derivative test alone gives no decision. However the first derivative is negative to the left and positive to the right, proving a minimum at (2, 1). Use another permitted test when none is specified, rather than labelling every zero second derivative a maximum or minimum.

A local maximum need not be the greatest value over the whole domain. The cubic example grows without bound to the right even though it has a local maximum. On a closed model interval, compare stationary values with both endpoint values to find the greatest or least permitted result. Points of inflexion are outside this section's stated requirements; the focus here is stationary points, justified extrema and relevant graph features.

Pause and explain

At a stationary point f″(a) = 0. What can this test alone conclude?

Put the idea to work

Worked example

Locate and classify the stationary points of y = x³ − 6x² + 9x + 1.

Show the worked solution

Set y′ = 3(x − 1)(x − 3) = 0, giving x = 1 or 3.
The original curve gives y = 5 and 1 respectively, so the points are (1, 5) and (3, 1).
Use y″ = 6x − 12: y″(1) = −6 gives a local maximum and y″(3) = 6 gives a local minimum; the derivative signs confirm both.

Answer Local maximum (1, 5); local minimum (3, 1).

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

Find and classify the stationary point of y = x² − 4x + 7.

Give me a hint

Solve the first derivative, find the curve height and then use the second derivative.

Compare my reasoning

y′ = 2x − 4 is zero at x = 2.
The original curve gives y(2) = 3, so the point is (2, 3).
y″ = 2 > 0, proving a minimum; completing the square to (x − 2)² + 3 independently confirms it.

Minimum at (2, 3).

Look for these in your work

Find both stationary coordinates.
Justify the nature with a valid positive second derivative.

2 · Independent

Find and classify the stationary point of y = (x − 2)⁴ + 1 without making a conclusion from its zero second derivative alone.

Give me a hint

A first-derivative sign test works even when the second test is inconclusive.

Compare my reasoning

y′ = 4(x − 2)³ is zero only at x = 2, where y = 1.
y″ = 12(x − 2)² is zero there, so that test alone is inconclusive.
The first derivative is negative for x < 2 and positive for x > 2, proving a minimum at (2, 1).

Minimum at (2, 1), by a negative-to-positive first-derivative sign change.

Look for these in your work

Recognise an inconclusive test rather than treating it as a classification.
Use both neighbouring derivative signs to justify the minimum.

3 · Transfer

A supplied output model is R(x) = −x² + 10x + 4 for 0 ≤ x ≤ 8. Find its greatest value on the permitted interval and justify it.

Give me a hint

Find any interior stationary input, classify it and compare with both included endpoints.

Compare my reasoning

R′(x) = −2x + 10 vanishes at x = 5, within the domain; R″ = −2 < 0 gives a local maximum.
Evaluate R(5) = 29, R(0) = 4 and R(8) = 20.
The model increases before 5 and decreases after it; together with the endpoint comparison, its greatest permitted value is 29 at x = 5.

Greatest permitted value 29 at x = 5; endpoint values 4 and 20 are smaller.

Look for these in your work

Retain only stationary candidates in the permitted domain.
Compare the interior value with both closed endpoints for a whole-interval conclusion.

Common mistakes

A stationary-point answer needs both coordinates and a justified nature.
A zero second derivative makes that test inconclusive; it does not prove the point's nature.
A constrained optimum must include permitted endpoint comparisons.

Recall without your notes

What can you explain now?

For y = (x − 2)⁴ + 1, classify its stationary point when the second derivative is zero there.

Compare with the explanation

Minimum at (2, 1), justified by a first-derivative change from negative to positive.

The second-derivative test is inconclusive, but 4(x − 2)³ has the required sign change.

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Make it stick

Describe the full stationary-point routine from f′ = 0 to a labelled sketch. Explain what to do when f″ = 0 and when a model has closed endpoints.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.