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

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

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

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

5. 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
x = 2
x = 5 or x = – 1
= – 2 x y
y = 5 + 4x – x2 2 9
The stationary point is a maximum 5 –1
y = 2 – 22
= 9
There is a maximum point at (2, 9)

6. 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
y = 2 + 3x2 – x3
6x – 3x2 = 0 y x
maximum (2,6) 3x
= 0
(2 – x)
x = 0 or x = 2
minimum (0, 2)
= 6
– 6x
When x = 0 this is positive
When x = 2, is negative
and y = 2
and y = – 8
= 6
Minimum point
(0, 2)
Maximum point
(2, 6)

7. To sketch y = x4 – 4
= 4x3
When x = 0, y = – 4
At stationary points
= 0
When y = 0,
4x3 = 0
x4 – 4 = 0
x4 = 4
x = 0
y = – 4
x2 = 2
x = ± 2
= 12x2
= 0 x y
y = x4 – 4
Gradient before x = 0 is
negative
Gradient after x = 0 is
positive Ö2
– 4
– Ö2
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?