1. Test for maximum, minimum and points of inflexion
1. Find stationary point a : f ' (a) = 0
2. Study the sign of f ' to the right and left of a
+
+
+
maximum
­
f ' (a)=0
horizontal
point of inflexion
­
­ f ' (a)=0f ' (a)=0
+­
minimum
f ' (a)=0

2. Find the coordinates of the stationary points on the
curve y= x4
­ 4 x3
and determine their nature.
Example 5:

4. Find the stationary points for the curve
and determine their nature.
Sketch the function.
Example 6:

6. Find turning points for .
Sketch the curve.
Example 7:

7. For the curve y = x3
+ x2
find the stationary
points and sketch the curve.
Example 8:

8. y = x3
+ x2

9. By the end of the lesson you will be able to:
• Use the second derivative to test the nature of a
stationary point and/or point of inflexion.

10.
•
•
A
B
f is concave upwards
gradient is increasing
f ' is increasing
f '' > 0
If a function is
increasing then its
derivative is positive
Concavity

11.
f is concave downwards
gradient is decreasing
f ' is decreasing
f '' < 0
If a function is
decreasing then its
derivative is negative
•
• A
B

12. f is concave upwards f '' > 0
f is concave downwards f '' < 0

13. The second derivative and turning points
•
•
B
A
at A:
f ' (a)= 0
f is concave up. f ''(a) > 0
f '(a) = 0
A is a minimum point
f ' (b)= 0
f is concave down.
f '(b) = 0
f ''(b) < 0
B is a maximum point
at B:

14. P is a minimum point
at P :
f '(x) = 0
f ''(x) > 0
P is a maximum point
at P :
f '(x) = 0
f ''(x)< 0

15. P
•
A point of inflexion is when the curve changes
from concave down to concave up or vice versa.
P is a point of inflexion
•
P
at P:
f ''(x) = 0
f '' changes sign
If f ' is also zero then P is a horizontal point
of inflexion.

16. Find the points of inflection of
f (x) = x4
­ 4 x3
+ 5

18. Find the stationary points for the curve
y= 4 + 3x ­ x3
and determine their nature. Draw a
sketch of the function.

19. y= 4 + 3x ­ x3

20. Find the point of inflexion on the curve
y = 2 x 3
+ 3 x 2
+ 6 x ­ 7 and sketch its graph

21. y = 2 x 3
+ 3 x 2
+ 6 x ­ 7

22. For the curve y = x3
+ x2
find the stationary points
and point of inflexion.

23. y = x3
+ x2

24. Find turning points for y =x3
­ 3x2
­ 9x +2. Sketch
the curve.

25. y =x3
­ 3x2
­ 9x +2

26. Test for maximum, minimum and points of inflexion
Method 1: First derivative Sign test
1. Find a : f ' (a) = 0
2. Study the sign of f ' to the right
and left of a
++ ­ ­
maximum minimum
+
+
­
­
f ' (a)=0
horizontal
point of inflexion
Method 2 : Second derivative test
1. Find a : f ' (a) = 0
2. Study the sign of f ''(a)
f '' (a) <0
f '' (a) = 0
and
f '' changes
sign at a
f ' (a)=0
f ' (a)=0
f ' (a)=0
f '' (a) >0 minimum
maximum
point of inflexion

27. http://www.mindomo.com/view.htm?m=7af63ca6ad1b462fa05b7762859f45d3
Derivative puzzle 1
Big derivative puzzle