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
    The curve CC has equation

    y=x2e−xy = x^2 e^{-x}.
    (a) Show that

    d2ydx2=(x2−4x+2)e−x\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = (x^2 - 4x + 2)e^{-x}. (3 marks)
    (b) Hence find the exact xx-coordinates of the points of inflection of CC. Fully justify your answer. (4 marks)
    Level 4 · 7 marksTry it