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.

Start practising Read the lesson

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

DifficultyAllLevel 2Level 3Level 4
  1. 1
    Find the set of values of xx for which the curve y=x3−6x2+5y = x^3 - 6x^2 + 5 is convex.
    Level 2 · 3 marksTry it
  2. 2
    Find the set of values of xx for which the curve y=x3−12xy = x^{3} - 12x is convex. (2 marks)
    Level 2 · 2 marksTry it
  3. 3
    f(x)=x3−3x2−24x+7\text{f}(x) = x^3 - 3x^2 - 24x + 7
    (a) Find f′′(x)\text{f}''(x)

    (2 marks)
    (b) (i) Solve f′′(x)=0\text{f}''(x) = 0

    (ii) Hence find the values of xx for which the curve with equation y=f(x)y = \text{f}(x) is concave.

    (2 marks)
    Level 2 · 4 marksTry it