The document discusses concavity, turning points, and inflection points of functions. It states that the second derivative measures concavity, known as change in slope. If the second derivative is positive, the curve is concave up, if negative it is concave down, and if zero there may be a point of inflection. Turning points occur at stationary points, where the first derivative is zero. If the second derivative is positive at a turning point, it is a minimum, and if negative, it is a maximum. A point of inflection is where the concavity changes, which can be seen by checking the sign of the second derivative on either side of the point.

1. Concavity

2. Concavity
The second deriviative measures the change in slope with respect to x,
this is known as concavity

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

13. Turning Points

14. Turning Points
All turning points are stationary points.

15. Turning Points
All turning points are stationary points.
If f  x   0, minimum turning point

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

24. d2y
when x  1, 2  6 1  2
dx
 4  0

25. d2y
when x  1, 2  6 1  2
dx
 4  0
 - 1,2 is a maximum turning point

26. d2y
when x  1, 2  6 1  2
dx
 4  0
 - 1,2 is a maximum turning point
1 d2y 1
when x  , 2  6   2
3 dx  3
40

27. d2y
when x  1, 2  6 1  2
dx
 4  0
 - 1,2 is a maximum turning point
1 d2y 1
when x  , 2  6   2
3 dx  3
40
 1 22 
  ,  is a minimum turning point
 3 27 

28. Inflection Points

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

40. Alternative Way of Finding
Inflection Points

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

44. d3y
e.g. 3  24
dx

45. d3y
e.g. 3  24
dx
1 d3y
when x   , 3  24  0
2 dx

46. d3y
e.g. 3  24
dx
1 d3y
when x   , 3  24  0
2 dx
 there is a change in concavity

47. d3y
e.g. 3  24
dx
1 d3y
when x   , 3  24  0
2 dx
 there is a change in concavity
 1 
   ,3  is a point of inflection
 2 

48. d3y
e.g. 3  24
dx
1 d3y
when x   , 3  24  0
2 dx
 there is a change in concavity
 1 
   ,3  is a point of inflection
 2 
Exercise 10E; 1, 2bc, 3, 6ac,
7bd, 8, 10, 12, 14, 16, 18