Stationary points

SPs

1. Block 1
Stationary Points

2. What is to be learned?
•How to use differentiation to identify SPs
•What SPs are!!!!!!
•What the difference between a stationary
point and a stationary value
•What SVs are
•How to find SP/SVs
•How to find the nature of SP/SVs

3. m ~ ½
m ~ 3

4. m ~ ½
m ~ 3
m ~ ½

5. m ~ ½
m ~ 3
m ~ ½

6. m ~ ½
m ~ 3
m ~ ½
m = 0

7. m ~ ½
m ~ 3
m ~ ½
m = 0

8. At Turning Points the gradient = 0
Turning points are known as stationary
points (or values)
(SPs or SVs)
So for SPsthe derivative = 0

9. Ex y = x
2
– 8x + 10 SPs?
For SPs
dy
/
dx
= 0

10. Ex y = x
2
– 8x + 10 SPs?
For SPs
dy
/
dx
= 0

11. Ex y = x
2
– 8x + 10 SPs?
For SPs
dy
/
dx
= 0
dy
/
dx
= 2x – 8
2x – 8 = 0
2x = 8
x = 4
y?

12. Ex y = x
2
– 8x + 10 SPs?
For SPs
dy
/
dx
= 0
dy
/
dx
= 2x – 8
2x – 8 = 0
2x = 8
x = 4
y?
y = 4
2
– 8(4) + 10
= - 6
SP is (4 , -6)
at x = 4

13. Stationary Points
•Max TPs, Min TPs or…….
•For SPs gradient = 0
•For SPs
dy
/
dx
= 0
Tactics
Find derivative
Solve equation → x
y → sub x into original equation
= 0

14. Ex y = x
2
+ 4x – 11 SPs?
For SPs
dy
/
dx
= 0
dy
/
dx
= 2x + 4
2x + 4 = 0
2x = -4
x = -2
y?
y = (-2)
2
+ 4(-2) – 11
= -15
SP is (-2 , -15)
at x = -2

15. Ex y = 20x – 2x
2
SP?
For SPs
dy
/
dx
= 0
dy
/
dx
= 20 – 4x
20 – 4x = 0
x = 5
y?
y = 20(5) – 2(5)
2

= 50
SP is (5 , 50)
at x = 5
Key Question

16. Ex y = 2x
3
– 6x
2
+ 10 SPs?
For SPs
dy
/
dx
= 0
dy
/
dx
= 6x
2
– 12x
6x
2
– 12x = 0
6x(x – 2) = 0
6x = 0 or x – 2 = 0
x = 0 or x = 2
Quadratic Equation
Factorise
x = 0 y = 10
x = 2 y =2(2)
3
– 6(2)
2
+ 10
= 2
(0 , 10) and (2 , 2)
Two SPs!!!→ Need two y values

17. Ex y =
1
/
3
x
3
– 2x
2
– 12xSPs?
For SPs
dy
/
dx
= 0
dy
/
dx
= x
2
– 4x – 12
x
2
– 4x – 12 = 0
(x – 6)(x + 2) = 0
x–6= 0 or x+2 = 0
x = 6 or x = -2
Quadratic Equation
Factorise
More Than One SP
Two SPs!!!
→ Need two y values

18. Ex y =
1
/
3
x
3
– 2x
2
– 12xSPs?
For SPs
dy
/
dx
= 0
dy
/
dx
= x
2
– 4x – 12
x
2
– 4x – 12 = 0
(x – 6)(x + 2) = 0
x–6= 0 or x+2 = 0
x = 6 or x = -2
More Than One SP
x = 6
y =
1
/
3
(6)
3
– 2(6)
2
– 12(6)
= - 72
x = -2
y =
1
/
3
(-2)
3
– 2(-2)
2
– 12(-2)
= 13
2
/
3
SPs are (-2 , 13
2
/
3) and (6 , -72)
Two SPs!!!
→ Need two y values

19. Find the SPs.
1. y =
1
/
3
x
3
– 3x
2
+ 8x
2.y = 2x
3
– 6x
2

3.y = x
3
– 3x
2
– 24x + 2
4.y = x
3
– 48x
(0 , 0) and (2 , -8)
(4 , -78) and (-2 , 30)
(4 , -128) and (-4 , 128)
Key
Question

20. Ex y =
1
/
3
x
3
– 3x
2
+ 8x SPs?
For SPs
dy
/
dx
= 0
dy
/
dx
= x
2
– 6x + 8
x
2
– 6x + 8 = 0
(x – 4)(x – 2 ) = 0
x–4= 0 or x–2 = 0
x = 4 or x = 2
x = 4
y =
1
/
3
(4)
3
– 3(4)
2
+ 8(4))
= 5
1
/
3
x = 2
y =
1
/
3
(2)
3
– 3(2)
2
+ 8(2)
= 6
2
/
3
SPs are (4 , 5
1
/
3) and (2 , 6
2
/
3)
Key
Question

21. What is to be learned?
•How to use differentiation to identify SPs
•What SPs are!!!!!!
•What the difference between a stationary
point and a stationary value
•What SVs are
•How to find SP/SVs
•How to find the nature of SP/SVs

22. Finding the nature

23. m negative
m positive m = 0
for nature need to know gradient
just before and after SP
Use the derivative!

24. Making a nature table

25. Making a Nature Table
y = x
2
– 8x + 10
For SPs
dy
/
dx
= 0
dy
/
dx
= 2x – 8
2x – 8 = 0
2x = 8
x = 4
y?
at x = 4y = 4
2
– 8(4) + 10
= - 6
SP is (4 , -6)

26. Making a Nature Table
y = x
2
– 8x + 10
dy
/
dx
= 2x – 8

SP is (4 , -6)
x
4
dydy
//
dxdx = 2x – 8 = 2x – 8 0
3 5
- +
Slope
Min TP
at (4 , -6)

27. Making a Nature Table
y = 2x
3
– 6x
2
+ 10
dy
/
dx
= 6x
2
– 12x

SPs (0 , 10) and (2 , 2)
x
0
dydy
//
dxdx = 6x = 6x
22
– 12x – 12x 0
-1 1
+ -
Slope
Max TP
at (0 , 10)
2
3
+0
Min TP
at (2 , 2)

28. Nature Table
•Used to find nature of SPs
•Show gradient just before and after SPs
•Use derivative

29. Making a Nature Table
y =
1
/
3
x
3
–– 2x
2
–– 12x
dy
/
dx
= x
2
– 4x - 12

SPs (-2, 13
2
/
3
) and (6 , -72)
x
-2
dydy
//
dxdx = x = x
22
– 4x – 4x –– 12 12 0
-3 0
+ -
Slope
Max TP
at (-2, 13
2
/
3)
6
7
+0
Min TP
at (6 , -72)

30. Making a Nature Table
y =
1
/
3
x
3
–– 2x
2
–– 12x
dy
/
dx
= x
2
– 4x – 12

SPs (-2, 13
2
/
3
) and (6 , -72)
x
-2
dydy
//
dxdx = x = x
22
– 4x – 12 – 4x – 12 0
-3 0
+ -
Slope
Max TP
at (-2, 13
2
/
3)
6
7
+0
Min TP
at (6 , -72)

31. Making a Nature Table
y =
1
/
3
x
3
–– 2x
2
–– 12x
dy
/
dx
= x
2
– 4x –– 12

SPs (-2, 13
2
/
3
) and (6 , -72)
x
-2
dydy
//
dxdx = (x = (x –– 6)(x + 2) 6)(x + 2) 0
-3 0
= +
Slope
Max TP
at (-2, 13
2
/
3)
6
7
0
Min TP
at (6 , -72)
= - = +
- X - - X + + X +

32. Making a Nature Table
y =
1
/
3
x
3
- 2x
2
– 12x
dy
/
dx
= x
2
– 4x – 12

SPs (-2, 13
2
/
3
) and (6 , -72)
x
-2
dydy
//
dxdx = x = x
22
– 4x – 4x –– 12 12 0
-3 0
+ -
Slope
Max TP
at (-2, 13
2
/
3)
6
7
+0
Min TP
at (6 , -72)
*
*could use
factorised
derivative
(( (x (x –– 6)(x + 2) 6)(x + 2) ))

33. Find Stationary Point and Nature for
y = 12x – 3x
2
For SPs
dy
/
dx
= 0
dy
/
dx
= 12 – 6x
12 – 6x = 0
x = 2
y = 12(2) – 3(2)
2

= 12
SP is (2 , 12)
at x = 2
Key Question
x
2
dydy
//
dxdx = 12 – 6x = 12 – 6x
0
1 3
+ -
Slope Max TP
at (2 , 12)