Nuffield FreeStanding Mathematics Activity Stationary points Maximum point

Nuffield Free-Standing Mathematics Activity Stationary points Maximum point Minimum point © Nuffield Foundation 2011

Stationary points Maximum point Think about What happens to the gradient as the curve passes through • a maximum point? • a minimum point? • a point of inflexion? point of inflexion Minimum point

Stationary points At a maximum point =0 0 + – is negative At a minimum point =0 is positive + – 0

Stationary points At some stationary points =0 and + 0 + =0 These are: points of inflexion – Note is also 0 at some maximum and minimum points 0 –

To sketch y = 5 + 4 x – x 2 When x = 0, y = 5 = 4 – 2 x When y = 0, At stationary points =0 2 x = 4 ( 5 - x)(1+ x) = 0 x=2 =– 2 The stationary point is a maximum y = 5 + 4 2 – 5 + 4 x – x 2 = 0 22 =9 There is a maximum point at (2, 9) x=5 or x = – 1 y y = 5 + 4 x – x 2 9 5 – 1 0 2 5 x

To sketch y = 2 + 3 x 2 – x 3 = 6 x – 3 x 2 When y = 0, 2 + 3 x 2 – x 3 = 0 At stationary points this is 0 6 x – 3 x 2 = 0 3 x (2 – x) = 0 x=0 or When x = 0, y = 2 x=2 = 6 – 6 x When x = 0 this is positive and y = 2 Minimum point (0, 2) y = 2 + 3 x 2 – x 3 maximum y (2, 6) minimum (0, 2) x 0 When x = 2, is negative and y = 2 + 12 – 8 = 6 Maximum point (2, 6)

To sketch y = x 4 – 4 = 4 x 3 When x = 0, y = – 4 At stationary points =0 4 x 3 = 0 x=0 y=– 4 When y = 0, x 4 – 4 = 0 x 4 = 4 x 2 = 2 x = ± 2 y y = x 4 – 4 0 Ö 2 = 12 x 2 = 0 Gradient before x = 0 is negative Gradient after x = 0 is positive – Ö 2 There is a minimum point at (0, – 4) – 4 x

Stationary points Reflect on your work How does finding stationary points help you to sketch a curve? Is it possible to know how many stationary points there are by just looking at the function? What other information is it useful to find before you attempt to sketch a curve?