A-level · Differentiation

Stationary points (maxima and minima)Exam-style questions

Finding where a curve turns, and using it to find the greatest or least value of something. What a stationary point is and why dy/dx = 0 finds it; full coordinates and which roots to keep; telling a maximum from a minimum with the second derivative; what to do when the second derivative is zero; where a curve is increasing or decreasing; and optimisation questions in context.

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25 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve $C$ has equation

$y = 2x^{\frac{3}{2}} + \dfrac{96}{\sqrt{x}} - 15 \qquad x > 0$
(a) Find $\dfrac{dy}{dx}$ , giving each term in simplest form.

(2 marks)
(b) Hence find the $x$ coordinate of the stationary point of $C$ .

(2 marks)
(c) (i) Find $\dfrac{d^2y}{dx^2}$ , giving each term in simplest form.

(ii) Hence determine the nature of the stationary point of $C$ , giving a reason for your answer.

(2 marks)
(d) State the range of values of $x$ for which $y$ is decreasing.

(1 mark)
Level 3 · 7 marksTry it
2
In this question you must show detailed reasoning.

The curve $C$ is defined by $y = x^3 - 6x^2 + 15x - 2$ .
(a) Show that there are no stationary points on $C$ .

(2 marks)
(b) Show that $C$ has only one point of inflection.

(2 marks)
Level 3 · 4 marksTry it
3
The curve $C$ has equation $y = 2x^3 - 9x^2 + 12x$ .
(a) Find the coordinates of the two stationary points of $C$ . (4 marks)
(b) Determine the nature of each stationary point. (3 marks)
Level 3 · 7 marksTry it
4
Figure 5 shows an open-topped storage tank in the shape of a cuboid. The base is a square of side $x$ metres and the height of the tank is $h$ metres.Figure 5 (not accurately drawn)

The tank must hold exactly $32$ m³ of liquid. The base is made from a reinforced material costing £ $15$ per m², and the four vertical sides are made from a lighter material costing £ $6$ per m².
(a) Show that the total cost, £ $C$ , of the material used to make the tank is given by

$C = 15x^2 + \dfrac{768}{x}$

(4 marks)
(b) Use calculus to find the value of $x$ for which $C$ is a minimum, giving your answer to $3$ significant figures, and prove that your value gives a minimum. (5 marks)
(c) Hence find the minimum cost of the material, to the nearest pound. (1 mark)
Level 4 · 10 marksTry it
5
The curve $C$ has equation $y = x^3 e^{-x}$ .
(a) Show that $\dfrac{\mathrm{d}y}{\mathrm{d}x} = x^2 e^{-x}(3 - x)$ . (2 marks)
(b) Find the coordinates of the stationary points of $C$ . (2 marks)
(c) Determine the nature of each stationary point. (3 marks)
Level 4 · 7 marksTry it

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