Academic Skills Advice
Differentiation Refresher Sheet 2
Differentiation is about gradients on a curve.
Stationary Points:
You need to be able to find a stationary point on a curve and decide whether it is a turning
point (maximum or minimum) or a point of inflexion.
When
𝑑𝑦
𝑑𝑥
= 0 the gradient is zero and so you have a stationary point.
Maximum turning point
Point of inflexion
Minimum turning point
2nd Order Differentiation:
The 2nd differential measures the change in gradient and can help you decide whether a
stationary point is a maximum, minimum or point of inflexion. To find the 2nd differential
you need to differentiate again.
If
If
If
𝑑𝑦
𝑑𝑥
𝑑𝑦
𝑑𝑥
𝑑𝑦
𝑑𝑥
= 0 and
= 0 and
= 0 and
𝑑2 𝑦
𝑑𝑥 2
𝑑2 𝑦
𝑑𝑥 2
𝑑2 𝑦
𝑑𝑥 2
>0
then you have a minimum (positive is minimum).
<0
then you have a maximum (negative is maximum).
=0
then it could be maximum, minimum or point of inflexion.
In the last case – differentiate again:
If
𝑑3 𝑦
𝑑𝑥 3
≠0
then it is a point of inflexion.
Otherwise (or as an alternative to the above) you need to find the gradient at either side of
the stationary point to decide whether it is a maximum or minimum.
© H Jackson 2008 / Academic Skills
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Examples:

Find the stationary point on the curve 𝒚 = 𝒙𝟐 − 𝟒𝒙 + 𝟐:
𝑑𝑦
𝑑𝑥
= 2𝑥 − 4 (we need to put this equal to zero to find the stationary point)
2𝑥 − 4 = 0
2𝑥 = 4
𝑥=2
(we know that the stationary point is at 𝑥 = 2.)
Next find the value of 𝑦 when 𝑥 = 2
𝑦 = 𝑥 2 − 4𝑥 + 2
𝑦 = 22 − 4 × 2 + 2
𝑦 = −2
So the stationary point is at (2, -2)
Next decide whether the stationary point is a maximum, minimum or point of inflexion:
𝑑2 𝑦
𝑑𝑥 2

=2
This is positive (>0) so the turning point must be a minimum.
Find the stationary points on the curve 𝒇(𝒙) = 𝒙𝟑 − 𝟑𝒙𝟐 + 𝟒:
𝑓 ′ (𝑥) = 3𝑥 2 − 6𝑥
3𝑥 2 − 6𝑥 = 0
3𝑥(𝑥 − 2) = 0
𝑥 = 0 𝑜𝑟 𝑥 = 2
(we need to put this equal to zero to find the stationary point)
(we know that the stationary points are at 𝑥 = 0 𝑎𝑛𝑑 𝑥 = 2.)
Next find the 𝑦 values:
𝑤ℎ𝑒𝑛 𝑥 = 0, 𝑦 = 4
𝑤ℎ𝑒𝑛 𝑥 = 2, 𝑦 = 0
stationary point is at (0,4)
stationary point is at (2,0)
Next decide whether the stationary points are maximum, minimum or point of inflexion:
𝑓 ′′ (𝑥) = 6𝑥 − 6
When 𝑥 = 0, 𝑓 ′′ (𝑥) = −6
When 𝑥 = 2, 𝑓 ′′ (𝑥) = +6

(maximum)
(minimum)
Find the stationary point on the curve 𝒚 = 𝒙𝟑 :
𝑑𝑦
𝑑𝑥
= 3𝑥 2
3𝑥 2 = 0
𝑑2 𝑦
𝑑𝑥 2
= 6𝑥
𝑥=0
Stationary point is at (0, 0)
This is 0 when 𝑥 = 0 so could be a max, min or point of inflexion.
Differentiate again to decide:
𝑑3 𝑦
𝑑𝑥 3
=6
The 3rd differential is not 0 so it is a point of inflexion.
© H Jackson 2008 / Academic Skills
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Practice Questions:
Find the stationary points and determine whether they are maximum, minimum or points of
inflexion for the following curves:
1)
𝒚 = −2𝒙𝟐 + 8𝒙 − 𝟕
2)
𝒚 = 3𝒙𝟐 − 12𝒙 + 𝟐
3)
𝒚 = 𝒙𝟐 + 8𝒙 − 7
4)
𝒇(𝒙) = 3 𝒙3 + 3𝒙𝟐 + 9𝒙 − 𝟑
5)
𝒚 = 𝟑 𝒙𝟑 − 𝒙𝟐 − 𝟏𝟐 𝒙 + 𝟏
6)
𝒇(𝒙) = 𝒙𝟑 − 𝟏𝟐𝒙
7)
𝒚 = −𝒙𝟑
8)
𝒇(𝒙) = 𝒙𝟑 − 𝟔𝒙𝟐 + 𝟏𝟐𝒙
9)
𝒚 = 𝒙𝟑 −
1
(Hint: factorise)
𝟐
𝟑
𝟐
𝒙𝟐 − 𝟏𝟖𝒙 + 𝟓
Challenge Activity:
10) Write 3 equations for curves of your own choice. Make sure that one curve has a
maximum, one has both a maximum and a minimum and one has a point of inflexion
and prove each one.
© H Jackson 2008 / Academic Skills
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