Edexcel IAL P2 Stationary Points: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

P2 differentiation uses $\frac{d y}{d x}$ to find where a function is increasing or decreasing and where it has stationary points, then the second derivative to decide their nature, and applies this to optimisation problems. At a stationary point $\frac{d y}{d x} = 0$ ; if $\frac{d^{2} y}{d x^{2}} < 0$ it is a maximum and if $\frac{d^{2} y}{d x^{2}} > 0$ a minimum.

P2 · Pure Mathematics 2
Calculator allowed
3 worked examples
33 practice questions
Tested in 4 of the last 4 papers

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Notes
Formulas
Worked examples
Mistakes
FAQ

P2 Stationary Points notes

Increasing and decreasing functions

$f$ is increasing where $f^{'} (x) > 0$ and decreasing where $f^{'} (x) < 0$ . For $f (x) = x^{3} - 3 x$ , $f^{'} (x) = 3 x^{2} - 3$ , so $f$ is increasing for $x < - 1$ or $x > 1$ and decreasing for $- 1 < x < 1$ .

Stationary points and their nature

Solve $\frac{d y}{d x} = 0$ , then find each $y$ -coordinate from the curve. On $y = x^{3} - 3 x$ the stationary points are $(- 1, 2)$ and $(1, - 2)$ . With $\frac{d^{2} y}{d x^{2}} = 6 x$ : at $x = - 1$ it is $- 6 < 0$ , a maximum; at $x = 1$ it is $6 > 0$ , a minimum. If the second derivative is zero, test the gradient either side instead.

Optimisation

Write the quantity to maximise or minimise in terms of one variable, differentiate, set to zero and check the nature. An open box is made from a 12 cm square sheet by cutting squares of side $x$ cm from the corners: $V = x (12 - 2 x)^{2}$ , and $\frac{d V}{d x} = 0$ gives $x = 2$ (the other root, $x = 6$ , gives no box), so the greatest volume is $128$ cm $^{3}$ .

Key formulas

Stationary points Learn it: $f^{'} (x) = 0; f^{″} < 0 max, f^{″} > 0 min$

In the formula booklet = printed in Pearson's IAL formulae booklet, so know where to find it. Learn it = not printed there, so learn it by heart. All P2 formulas on one page (free PDF) →

P2 Stationary Points worked examples

Each mark scheme is checked line by line before it is published here. M marks are for method, A marks for accurate answers that follow from them, B marks for independent results.

Worked example 1 · 4 marks · hard

Find the set of values of $x$ for which $f (x) = x^{4} - 8 x^{2}$ is an increasing function.

Mark scheme

M1 for finding the derivative $f^{'} (x) = 4 x^{3} - 16 x$ .
M1 for setting $f^{'} (x) > 0$ and factorising: $4 x (x^{2} - 4) > 0 ⟹ 4 x (x - 2) (x + 2) > 0$ .
M1 for finding the critical values $x = - 2, 0, 2$ and testing regions.
A1 for stating the correct range: $- 2 < x < 0$ or $x > 2$ .

4 steps

Worked example 2 · 4 marks · hard

Find the exact coordinates of the stationary point on the curve $y = \sqrt{x} - 2 x$ , for $x > 0$ .

Mark scheme

M1 for rewriting as $y = x^{0.5} - 2 x$ and differentiating: $\frac{d y}{d x} = 0.5 x^{- 0.5} - 2$ .
M1 for setting to zero and rearranging: $\frac{1}{2 \sqrt{x}} = 2 ⟹ \sqrt{x} = 0.25$ .
A1 for solving to get $x = 0.0625$ or $\frac{1}{16}$ .
A1 for finding the y-coordinate: $\sqrt{\frac{1}{16}} - 2 (\frac{1}{16}) = \frac{1}{4} - \frac{1}{8} = \frac{1}{8}$ , giving $(\frac{1}{16}, \frac{1}{8})$ .

4 steps

Worked example 3 · 4 marks · medium

Find the coordinates of the stationary points on the curve $y = 2 x^{3} - 15 x^{2} + 24 x + 6$ .

Mark scheme

M1 for differentiating to find $\frac{d y}{d x} = 6 x^{2} - 30 x + 24$ .
M1 for setting to $0$ and factorising: $6 (x^{2} - 5 x + 4) = 0 ⟹ 6 (x - 4) (x - 1) = 0$ .
A1 for finding the x-coordinates $x = 1$ and $x = 4$ .
A1 for finding the y-coordinates to state the full points: $(1, 17)$ and $(4, - 10)$ .

4 steps · 1 to fill in

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Common P2 stationary points and optimisation mistakes

Stopping at the x-coordinate. A stationary point is a point: give its y-coordinate too, from the curve's equation.
No reason for the nature. Write the value of $\frac{d^{2} y}{d x^{2}}$ and say "< 0 so maximum" (or > 0, minimum). A bare "maximum" may not get the mark.
Using the derivative's value as y. Substitute into $y$ , not into $\frac{d y}{d x}$ (which is 0 there).
Impossible answers in context. Check the domain: a length cannot be negative and a box cannot have zero width. Reject the values that make no sense.

Every P2 mistake from the examiner reports, with the paper it came from →

On your calculator: how to find a numerical derivative and definite integral. A calculator is allowed in every IAL unit.

P2 Stationary Points FAQ

How do I tell if a stationary point is a maximum or a minimum?

Work out the second derivative there: negative means a maximum, positive a minimum. If it is zero, look at the sign of the gradient just either side of the point.

What does increasing function mean?

As x increases, f(x) increases, so f′(x) > 0 on that interval. Questions often ask for the set of values of x for which a function is increasing.

Is a calculator allowed in IAL P2?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P2 included. Method marks are for the working you write down, so show it.

How long is the IAL P2 exam?

Each IAL maths unit, P2 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does stationary points and optimisation come up in P2?

In our mapping of the 4 Pearson P2 papers from June 2024 to June 2025, stationary points and optimisation was tested in 4 of them (5 questions in all). Any topic can be examined in any series.

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More P2: Sequences and Series · Logarithms and Exponentials · Circles · Further Trigonometry · Algebraic Methods · Proof · The Binomial Expansion · Integration (definite integrals and area)

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Other units: IAL P1 · IAL P3 · IAL P4 · IAL S1 · IAL M1

Edexcel IAL P2 Stationary Points: exam questions and answers, notes and worked examples

Try it first

P2 Stationary Points notes

Increasing and decreasing functions

Stationary points and their nature

Optimisation

Key formulas

P2 Stationary Points worked examples

Worked example 1 · 4 marks · hard

Mark scheme

Worked example 2 · 4 marks · hard

Mark scheme

Worked example 3 · 4 marks · medium

Mark scheme

Common P2 stationary points and optimisation mistakes

P2 Stationary Points FAQ

How do I tell if a stationary point is a maximum or a minimum?

What does increasing function mean?

Is a calculator allowed in IAL P2?

How long is the IAL P2 exam?

How often does stationary points and optimisation come up in P2?

Next steps

Your P2 revision kit