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

2. Module C1 Module C2 AQA OCR Edexcel MEI/OCR
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3. The stationary points of a curve are the points where the gradient is zero
We may be able to determine the nature of a stationary point just by knowing the shape of a curve.
e.g.1 We know the curve has a minimum because the sign of the term ( positive ) tells us that the graph has the following shape

4. e.g.1 The cubic curve has 2 stationary points.
They are and
Plotting the stationary points and using our knowledge that a cubic is a continuous function
( we can draw it with a single stroke ) means we must get the following: x x
The y-intercept is also useful

5. e.g.1 The cubic curve has 2 stationary points.
They are and
Plotting the stationary points and using our knowledge that a cubic is a continuous function
( we can draw it with a single stroke ) means we must get the following: x

6. We may not know the shape of some functions, so we need to determine the nature of the stationary points by another method.
Using the 2nd derivative is usually the easiest method.

7. Distinguish between the max and the min.
e.g.2 Calculate the coordinates of the stationary points on the graph of where
Distinguish between the max and the min.
must be written in the form before we can differentiate
Solution:
For st. pts.
Multiply by :
this quadratic equation has no linear term so there is no need to factorize

8. N.B. The maximum has a smaller y -value than the minimum !
Calculate y-values at x = 1 and -1:
The stationary points are ( 1, 2 ) and (-1, -2)
To distinguish between the stationary points we need the 2nd derivative
is a min
is a max
N.B. The maximum has a smaller y -value than the minimum !
It’s interesting to see what the graph looks like.

9. x = 0 is infinite, so x = 0 (the y-axis) is an asymptote
So, we now have
x = 0 x
(min) x
(max)

10. “ x approaches infinity “
Also, as so
“ x approaches infinity “
“ approaches x “
“ approaches zero “
is also an asymptote
x = 0
y = x x
(min) x
(max)

11. We can now complete the curve.x
(max)
(min)
Asymptote, x = 0

12. Exercise
1. Find the stationary points on the curve
where
Determine the nature of the stationary points.
Ans:
is a maximum
is a minimum
The question didn’t ask for the graph but it looks like this:
The asymptotes are

14. 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.

15. Distinguish between the max and the min.
Solution:
e.g.2 Calculate the coordinates of the stationary points on the graph of where
Multiply by :
must be written in the form before we can differentiate
For st. pts.
this quadratic equation has no linear term so there is no need to factorize

16. N.B. The maximum has a smaller y -value than the minimum !
To distinguish between the stationary points we need the 2nd derivative
The stationary points are ( 1, 2 ) and (-1, -2)
is a min
is a max
Calculate y-values at x = 1 and -1: