Nuffield Free-Standing Mathematics Activity Stationary points

1. Nuffield Free-Standing Mathematics Activity Stationary points Maximum point Minimum point

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

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

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

5. y y = 5 + 4x – x2 9 5 –1 0 5 x 2 To sketch y = 5 + 4x – x2 When x = 0, y = 5 = 4 – 2x When y = 0, At stationary points = 0 5 + 4x – x2 = 0 2x= 4 ( - x)( + x) = 0 5 1 x= 2 x= 5 or x= – 1 = – 2 The stationary point is a maximum =9 y = 5 + 4 2 – 22 There is a maximum point at (2, 9)

6. y = 2 + 3x2 – x3 maximum(2,6) y 3x = 0 minimum (0, 2) x 0 Maximum point (2, 6) (0, 2) Minimum point To sketch y = 2 + 3x2 – x3 When x = 0, y = 2 = 6x – 3x2 When y = 0, 2 + 3x2 – x3 = 0 At stationary points this is 0 6x – 3x2 = 0 (2 – x) x = 0 or x= 2 = 6 – 6x When x= 0 this is positive When x= 2, is negative andy= 2 andy= 2 + 12 – 8 = 6

7. = 0 y y = x4 – 4 0 x – Ö2 Ö2 – 4 To sketch y = x4 –4 = 4x3 When x = 0, y = – 4 At stationary points When y = 0, 4x3= 0 x4 – 4 = 0 x4 = 4 x= 0 y = – 4 x= ± 2 x2 = 2 = 0 = 12x2 negative Gradient before x = 0 is positive Gradient after x = 0 is There is a minimum point at (0, – 4)

8. 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?