3. Concavity
The second deriviative measures the change in slope with respect to x,
this is known as concavity
If f x 0, the curve is concave up
4. Concavity
The second deriviative measures the change in slope with respect to x,
this is known as concavity
If f x 0, the curve is concave up
If f x 0, the curve is concave down
5. Concavity
The second deriviative measures the change in slope with respect to x,
this is known as concavity
If f x 0, the curve is concave up
If f x 0, the curve is concave down
If f x 0, possible point of inflection
6. Concavity
The second deriviative measures the change in slope with respect to x,
this is known as concavity
If f x 0, the curve is concave up
If f x 0, the curve is concave down
If f x 0, possible point of inflection
e.g. By looking at the second derivative sketch y x 3 5 x 2 3x 2
7. Concavity
The second deriviative measures the change in slope with respect to x,
this is known as concavity
If f x 0, the curve is concave up
If f x 0, the curve is concave down
If f x 0, possible point of inflection
e.g. By looking at the second derivative sketch y x 3 5 x 2 3x 2
dy
3x 2 10 x 3
dx
8. Concavity
The second deriviative measures the change in slope with respect to x,
this is known as concavity
If f x 0, the curve is concave up
If f x 0, the curve is concave down
If f x 0, possible point of inflection
e.g. By looking at the second derivative sketch y x 3 5 x 2 3x 2
dy
3x 2 10 x 3
dx
d2y
2
6 x 10
dx
9. Concavity
The second deriviative measures the change in slope with respect to x,
this is known as concavity
If f x 0, the curve is concave up
If f x 0, the curve is concave down
If f x 0, possible point of inflection
e.g. By looking at the second derivative sketch y x 3 5 x 2 3x 2
dy
3x 2 10 x 3
dx
d2y
2
6 x 10
dx
d2y
Curve is concave up when 2 0
dx
10. Concavity
The second deriviative measures the change in slope with respect to x,
this is known as concavity
If f x 0, the curve is concave up
If f x 0, the curve is concave down
If f x 0, possible point of inflection
e.g. By looking at the second derivative sketch y x 3 5 x 2 3x 2
dy
3x 2 10 x 3
dx
d2y
2
6 x 10
dx
d2y
Curve is concave up when 2 0
dx
i.e. 6 x 10 0
5
x
3
11. Concavity
The second deriviative measures the change in slope with respect to x,
this is known as concavity
If f x 0, the curve is concave up
If f x 0, the curve is concave down
If f x 0, possible point of inflection
e.g. By looking at the second derivative sketch y x 3 5 x 2 3x 2
dy y
3x 2 10 x 3
dx
d2y
2
6 x 10
dx
d2y
Curve is concave up when 2 0 5 x
dx
i.e. 6 x 10 0 3
5
x
3
12. Concavity
The second deriviative measures the change in slope with respect to x,
this is known as concavity
If f x 0, the curve is concave up
If f x 0, the curve is concave down
If f x 0, possible point of inflection
e.g. By looking at the second derivative sketch y x 3 5 x 2 3x 2
dy y
3x 2 10 x 3
dx
d2y
2
6 x 10
dx
d2y
Curve is concave up when 2 0 5 x
dx
i.e. 6 x 10 0 3
5
x
3
16. Turning Points
All turning points are stationary points.
If f x 0, minimum turning point
If f x 0, maximum turning point
17. Turning Points
All turning points are stationary points.
If f x 0, minimum turning point
If f x 0, maximum turning point
e.g. Find the turning points of y x 3 x 2 x 1
18. Turning Points
All turning points are stationary points.
If f x 0, minimum turning point
If f x 0, maximum turning point
e.g. Find the turning points of y x 3 x 2 x 1
dy
3x 2 2 x 1
dx
19. Turning Points
All turning points are stationary points.
If f x 0, minimum turning point
If f x 0, maximum turning point
e.g. Find the turning points of y x 3 x 2 x 1
dy
3x 2 2 x 1
dx
d2y
2
6x 2
dx
20. Turning Points
All turning points are stationary points.
If f x 0, minimum turning point
If f x 0, maximum turning point
e.g. Find the turning points of y x 3 x 2 x 1
dy
3x 2 2 x 1
dx
d2y
2
6x 2
dx
dy
Stationary points occur when 0
dx
21. Turning Points
All turning points are stationary points.
If f x 0, minimum turning point
If f x 0, maximum turning point
e.g. Find the turning points of y x 3 x 2 x 1
dy
3x 2 2 x 1
dx
d2y
2
6x 2
dx
dy
Stationary points occur when 0
dx
i.e. 3x 2 2 x 1 0
22. Turning Points
All turning points are stationary points.
If f x 0, minimum turning point
If f x 0, maximum turning point
e.g. Find the turning points of y x 3 x 2 x 1
dy
3x 2 2 x 1
dx
d2y
2
6x 2
dx
dy
Stationary points occur when 0
dx
i.e. 3x 2 2 x 1 0
3x 1 x 1 0
23. Turning Points
All turning points are stationary points.
If f x 0, minimum turning point
If f x 0, maximum turning point
e.g. Find the turning points of y x 3 x 2 x 1
dy
3x 2 2 x 1
dx
d2y
2
6x 2
dx
dy
Stationary points occur when 0
dx
i.e. 3x 2 2 x 1 0
3x 1 x 1 0
1
x or x 1
3
29. Inflection Points
A point of inflection is where there is a change in concavity, to see if
there is a change, check either side of the point.
30. Inflection Points
A point of inflection is where there is a change in concavity, to see if
there is a change, check either side of the point.
e.g. Find the inflection point(s) of y 4 x 3 6 x 2 2
31. Inflection Points
A point of inflection is where there is a change in concavity, to see if
there is a change, check either side of the point.
e.g. Find the inflection point(s) of y 4 x 3 6 x 2 2
dy d2y
12 x 2 12 x 2
24 x 12
dx dx
32. Inflection Points
A point of inflection is where there is a change in concavity, to see if
there is a change, check either side of the point.
e.g. Find the inflection point(s) of y 4 x 3 6 x 2 2
dy d2y
12 x 2 12 x 2
24 x 12
dx dx 2
d y
Possible points of inflection occur when 2 0
dx
33. Inflection Points
A point of inflection is where there is a change in concavity, to see if
there is a change, check either side of the point.
e.g. Find the inflection point(s) of y 4 x 3 6 x 2 2
dy d2y
12 x 2 12 x 2
24 x 12
dx dx 2
d y
Possible points of inflection occur when 2 0
dx
i.e. 24 x 12 0
1
x
2
34. Inflection Points
A point of inflection is where there is a change in concavity, to see if
there is a change, check either side of the point.
e.g. Find the inflection point(s) of y 4 x 3 6 x 2 2
dy d2y
12 x 2 12 x 2
24 x 12
dx dx 2
d y
Possible points of inflection occur when 2 0
dx
i.e. 24 x 12 0 1 1 1
1 x
x 2 2 2
2
d2y 0
dx 2
35. Inflection Points
A point of inflection is where there is a change in concavity, to see if
there is a change, check either side of the point.
e.g. Find the inflection point(s) of y 4 x 3 6 x 2 2
dy d2y
12 x 2 12 x 2
24 x 12
dx dx 2
d y
Possible points of inflection occur when 2 0
dx
i.e. 24 x 12 0 1 1 1
1 x
x 2 2 2 (0)
2
d2y 0
(12)
dx 2
36. Inflection Points
A point of inflection is where there is a change in concavity, to see if
there is a change, check either side of the point.
e.g. Find the inflection point(s) of y 4 x 3 6 x 2 2
dy d2y
12 x 2 12 x 2
24 x 12
dx dx 2
d y
Possible points of inflection occur when 2 0
dx
i.e. 24 x 12 0 1 1 1
1 x
x 2 (-1) 2 2 (0)
2
d 2 y (-12) 0 (12)
dx 2
37. Inflection Points
A point of inflection is where there is a change in concavity, to see if
there is a change, check either side of the point.
e.g. Find the inflection point(s) of y 4 x 3 6 x 2 2
dy d2y
12 x 2 12 x 2
24 x 12
dx dx 2
d y
Possible points of inflection occur when 2 0
dx
i.e. 24 x 12 0 1 1 1
1 x
x 2 (-1) 2 2 (0)
2
there is a change in concavity d 2 y (-12) 0 (12)
dx 2
38. Inflection Points
A point of inflection is where there is a change in concavity, to see if
there is a change, check either side of the point.
e.g. Find the inflection point(s) of y 4 x 3 6 x 2 2
dy d2y
12 x 2 12 x 2
24 x 12
dx dx 2
d y
Possible points of inflection occur when 2 0
dx
i.e. 24 x 12 0 1 1 1
1 x
x 2 (-1) 2 2 (0)
2
there is a change in concavity d 2 y (-12) 0 (12)
1
,3 is a point of inflection
dx 2
2
39. Inflection Points
A point of inflection is where there is a change in concavity, to see if
there is a change, check either side of the point.
e.g. Find the inflection point(s) of y 4 x 3 6 x 2 2
dy d2y
12 x 2 12 x 2
24 x 12
dx dx 2
d y
Possible points of inflection occur when 2 0
dx
i.e. 24 x 12 0 1 1 1
1 x
x 2 (-1) 2 2 (0)
2
there is a change in concavity d 2 y (-12) 0 (12)
1
,3 is a point of inflection
dx 2
2
dy d2y d3y
Horizontal Point of Inflection; 0 2
0 3
0
dx dx dx
41. Alternative Way of Finding
Inflection Points
d2y
Possible points of inflection occur when 2 0
dx
42. Alternative Way of Finding
Inflection Points
d2y
Possible points of inflection occur when 2 0
dx
If the first non-zero derivative is of odd order,
d3y d5y d7y
i.e 3
0 or 5
0 or 7
0 etc
dx dx dx
then it is a point of inflection
43. Alternative Way of Finding
Inflection Points
d2y
Possible points of inflection occur when 2 0
dx
If the first non-zero derivative is of odd order,
d3y d5y d7y
i.e 3
0 or 5
0 or 7
0 etc
dx dx dx
then it is a point of inflection
If the first non-zero derivative is of even order,
d4y d6y d8y
i.e 4
0 or 6
0 or 8
0 etc
dx dx dx
then it is not a point of inflection