Critical points of a function occur when the derivative is equal to 0 or undefined. These include maximum and minimum turning points where the derivative is 0, horizontal inflection points where the derivative is 0, and cusp points where the derivative is undefined.

1. Critical Points

2. Critical Points
undefinedisor0occur whenpointsCritical 
dx
dy

3. Critical Points
undefinedisor0occur whenpointsCritical 
dx
dy
y
x

4. Critical Points
undefinedisor0occur whenpointsCritical 
dx
dy
y
x

5. Critical Points
undefinedisor0occur whenpointsCritical 
dx
dy
y
x
  pointturningmaximum,0 xf

6. Critical Points
undefinedisor0occur whenpointsCritical 
dx
dy
y
x
  pointturningmaximum,0 xf
  pointturningminimum,0 xf

7. y
x

8. y
x

9. y
x
  inflectionofpointhorizontal,0 xf

10. y
x
y
x
  inflectionofpointhorizontal,0 xf

11. y
x
y
x
  inflectionofpointhorizontal,0 xf
  tangentvertical,undefinedisxf 

12. y
x

13. y
x

14. y
x
 
pointminimum
continuousnotiscurvecusp"",undefinedisxf 

15. y
x
 
pointminimum
continuousnotiscurvecusp"",undefinedisxf 
Exercise 10C; 1, 2ace etc, 3, 5, 8, 10