Critical points occur when the derivative of a function is equal to zero or undefined. This includes local maxima and minima, where the derivative is zero. It also includes points of inflection, where the second derivative is zero, and cusp points where the derivative is undefined. The document provides examples of graphs illustrating these different types of critical points.

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