A-level · Differentiation

Convex, concave and points of inflectionExam-style questions

Using the second derivative to describe how a curve bends. What convex and concave mean and why f'' decides them; finding the intervals where a curve is convex or concave; points of inflection, where the bending changes; why f''(x) = 0 is not enough on its own; stationary and non-stationary inflections; and showing a curve is convex everywhere.

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

DifficultyAll Level 2 Level 3 Level 4

1
Find the coordinates of the point of inflection of the curve $y = x^3 - 3x^2 - 9x + 1$ .
Level 3 · 4 marksTry it
2
The curve $C$ has equation $y = x^4 - 4x^3$ . Find the coordinates of the points of inflection of $C$ , and the set of values of $x$ for which $C$ is convex.
Level 3 · 5 marksTry it
3
Find the set of values of $x$ for which the curve $y = 2x^{3} - 9x^{2} + 12x$ is concave. (3 marks)
Level 3 · 3 marksTry it
4
The function f is defined by

$\mathrm{f}(x) = e^{2x} + e^{-2x} - 4x^2 \quad \text{for } x \in \mathbb{R}$

Determine whether the curve with equation $y = \mathrm{f}(x)$ has a point of inflection at the point where $x = 0$

Fully justify your answer.

(4 marks)
Level 3 · 4 marksTry it
5
Show that the curve $y = x^3 + 2$ has a stationary point of inflection, and state its coordinates.
Level 3 · 4 marksTry it

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