1. “Teach A Level Maths” Vol. 1: AS Core Modules
13: Stationary Points
© Christine Crisp

2. Module C1 Module C2 AQA Edexcel OCR MEI/OCR
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3. The stationary points of a curve are the points where the gradient is zero
A local maximum x x
A local minimum
The word local is usually omitted and the points called maximum and minimum points.

4. e.g.1 Find the coordinates of the stationary points on the curve
Solution:
Tip: Watch out for common factors when finding stationary points. or
The stationary points are (3, -27) and ( -1, 5)

5. Exercises
Find the coordinates of the stationary points of the following functions 1. 2.
Solutions: 1.
Ans: St. pt. is ( 2, 1)

6. 2.
Solution:
Ans: St. pts. are ( 1, -6) and ( -2, 21 )

7. We need to be able to determine the nature of a stationary point ( whether it is a max or a min ). There are several ways of doing this. e.g.
On the left of a maximum, the gradient is positive
On the right of a maximum, the gradient is negative

8. So, for a max the gradients are
At the max
On the left of the max
On the right of the max
The opposite is true for a minimum
Calculating the gradients on the left and right of a stationary point tells us whether the point is a max or a min.

9. e. g. 2 Find the coordinates of the stationary point of the curve
e.g.2 Find the coordinates of the stationary point of the curve Is the point a max or min?
Solution:
Substitute in (1):
On the left of x = 2 e.g. at x = 1,
On the right of x = 2 e.g. at x = 3,
We have
is a min

10. Another method for determining the nature of a stationary point.
e.g.3 Consider
The gradient function is given by
At the max of
the gradient is 0
but the gradient of the gradient is negative.

11. Another method for determining the nature of a stationary point.
e.g.3 Consider
The gradient function is given by
At the min of
the gradient of the gradient is positive.
The notation for the gradient of the gradient is
“d 2 y by d x squared”

12. e.g.3 ( continued ) Find the stationary points on the curve and distinguish between the max and the min.
Solution:
Stationary points:
is called the 2nd derivative or
We now need to find the y-coordinates of the st. pts.

13. To distinguish between max and min we use the 2nd derivative, at the stationary points.
max at
At ,
min at

14. SUMMARY
To find stationary points, solve the equation
Determine the nature of the stationary points
either by finding the gradients on the left and right of the stationary points
maximum
minimum
or by finding the value of the 2nd derivative at the stationary points

15. Exercises
Find the coordinates of the stationary points of the following functions, determine the nature of each and sketch the functions. 1.
is a min.
is a max.
Ans. 2.
is a min.
is a max.
Ans.

17. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

18. The stationary points of a curve are the points where the gradient is zero
A local maximum
A local minimum x
The word local is usually omitted and the points called maximum and minimum points.
e.g.

19. e.g.1 Find the coordinates of the stationary points on the curve
Solution: or
The stationary points are (3, -27) and ( -1, 5)

20. For a max we have
The opposite is true for a minimum
At the max
On the right of the max
On the left of the max
Calculating the gradients on the left and right of a stationary point tells us whether the point is a max or a min.
Determining the nature of a Stationary Point

21. At the max of the gradient is 0, but the gradient of the gradient is negative.
The gradient function is given by
e.g. Consider
Another method for determining the nature of a stationary point.

22. The notation for the gradient of the gradient is
“d 2 y by d x squared”
At the min of
The gradient function is given by
the gradient of the gradient is positive.

23. The gradient of the gradient is called the 2nd derivative and is written as

24. e.g. Find the stationary points on the curve and distinguish between the max and the min.
Solution:
Stationary points: or
We now need to find the y-coordinates of the st. pts.

25. max at
min at
At ,
To distinguish between max and min we use the 2nd derivative,

26. SUMMARY
To find stationary points, solve the equation
maximum
minimum
Determine the nature of the stationary points
either by finding the gradients on the left and right of the stationary points
or by finding the value of the 2nd derivative at the stationary points