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