About this chapter
Pure 2 differentiation takes the gradient work of Pure 1 and asks what it tells you: where a curve turns, whether it is a maximum or a minimum, where a function is increasing, and how to make a box, a window or a sheet of card as large or as small as possible.
The chapter covers stationary points and their nature through the second derivative, increasing and decreasing functions, showing a function increases for every x, and optimisation problems where a constraint removes the second variable. The June 2023 examiners found calculus in context one of the hardest parts of that paper, so three of the ten questions here are set in context.
The ten questions
- Q1 (6 marks): the stationary points of a cubic and their nature from the second derivative.
- Q2 (4 marks): where a cubic is decreasing, as a quadratic inequality.
- Q3 (5 marks): a stationary point of a curve with a power of 3/2, and its nature with a reason.
- Q4 (4 marks): show a cubic is increasing for every real x by completing the square on f′(x).
- Q5 (5 marks): stationary points of a cubic, then a sketch using them.
- Q6 (8 marks): an open box folded from a 30 cm square: the volume, the value of x that makes it largest, and why it is a maximum.
- Q7 (8 marks): a closed box of fixed volume: the surface area in terms of x, and its minimum.
- Q8 (8 marks): a window of fixed perimeter made of a rectangle and a semicircle, and its largest possible area.
- Q9 (9 marks): a given stationary point fixes two constants, then the other stationary point and where the curve increases.
- Q10 (10 marks): stationary points of a cubic, where the tangent at the maximum meets the curve again, and when f(x) = k has three solutions.
Key skills tested
Stationary points. Solve dy/dx = 0, then substitute each x into y, not into dy/dx.
Nature of a stationary point. A negative second derivative means a maximum and a positive one means a minimum. Give the value and a conclusion.
Increasing and decreasing. A function increases where f′(x) is positive and decreases where it is negative, which usually means solving a quadratic inequality.
Increasing for every x. Complete the square on f′(x). If it becomes a(x+p)² + q with a and q both positive, f′(x) is always positive.
Powers of x first. Write roots and fractions as powers before differentiating. For example, 8/x² is 8 times x to the power −2.
Forming the expression. Use the constraint, such as a fixed volume, perimeter or area, to remove the second variable.
Maximum and minimum values. Solve the derivative equal to zero, then substitute back into the original expression and give the units.
Practical domain. Reject any value that would make a length zero or negative, and say why.
Using stationary points. They fix the shape of a sketch, and the values of k for which f(x) = k has a given number of solutions.
Worked example
Question 1 from this chapter. The curve C has equation y = 2x³ − 9x² + 12x + 1. (a) Find the coordinates of the stationary points. (b) Use the second derivative to determine their nature.
(a) dy/dx = 6x² − 18x + 12 = 6(x−1)(x−2), which is zero at x = 1 and x = 2. Substituting into y: at x = 1, y = 2 − 9 + 12 + 1 = 6; at x = 2, y = 16 − 36 + 24 + 1 = 5. The stationary points are (1, 6) and (2, 5).
(b) d²y/dx² = 12x − 18. At x = 1 it equals −6, which is negative, so (1, 6) is a maximum. At x = 2 it equals 6, which is positive, so (2, 5) is a minimum.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- June 2024, Q9(a): a stationary point on a curve with fractional powers, found after expanding a bracket, as in Q3.
- June 2023, Q7: calculus in a real context, with fractional powers and the second derivative used to confirm the nature of the point, the skills of Q6 to Q8.
Where marks are lost
- The wrong letter. In June 2023 many replaced t with another letter before differentiating, then differentiated with respect to the wrong one.
- Checking either side. When a question says to use calculus to find the nature of a point, testing the gradient on each side earned no credit in June 2023. Use the second derivative.
- Differentiating the wrong expression. The most common error in June 2023 was differentiating the equation found for the stationary point instead of finding the second derivative of the original function.
- Solving with fractional powers. In June 2024, equations with powers of ½ and 3/2 caused trouble. Taking out the common factor of √x was the most successful method.
Common questions
How do I show a point is a maximum?
Find the second derivative's value at that point, state that it is negative, and conclude that the point is a maximum.
Are stationary points in Pure 1 or Pure 2?
Pure 2. Pure 1 differentiation stops at tangents, normals and second derivatives.
What should the final answer to an optimisation question be?
The maximum or minimum value of the quantity asked for, with its units, not just the value of x.
Where this chapter leads
- Pure 2 Chapter 8: Integration: areas of regions bounded by the curves sketched here.
- Pure 3 Chapter 6: Differentiation: the chain, product and quotient rules.