1. Critical Points

2. Critical Points
dy
Critical points occur when = 0 or is undefined
dx

3. Critical Points
dy
Critical points occur when = 0 or is undefined
dx
y
x

4. Critical Points
dy
Critical points occur when = 0 or is undefined
dx
y
x

5. Critical Points
dy
Critical points occur when = 0 or is undefined
dx
f ′( x ) = 0, maximum turning point
y
x

6. Critical Points
dy
Critical points occur when = 0 or is undefined
dx
f ′( x ) = 0, maximum turning point
y
x
f ′( x ) = 0, minimum turning point

7. y
x

8. y
x

9. y
f ′( x ) = 0, horizontal point of inflection
x

10. y
f ′( x ) = 0, horizontal point of inflection
x
y
x

11. y
f ′( x ) = 0, horizontal point of inflection
x
y
x
f ′( x ) is undefined, vertical tangent

12. y
x

13. y
x

14. y
x
f ′( x ) is undefined, " cusp" curve is not continuous
minimum point

15. y
x
f ′( x ) is undefined, " cusp" curve is not continuous
minimum point
Exercise 10C; 1, 2ace etc, 3, 5, 8, 10