1. Extrema on an Interval 3.1 On the agenda: Defining Extrema
Relative and Absolute Extrema
Critical Numbers
HW: p # 1-5 Odd, 7-10, Odd,

2. Terminology
– Absolute (or global) min/max
The absolute highest or lowest value(s) in an interval
– Relative (or Local) min/max
Local to the values around it
Where b, c, and d are Critical Numbers (where f’(x) is either 0 or undefined)

3. Terminology cont… – Relative Maximum => Can be thought of as a hill
– Relative Minimum => Can be thought of as a valley
• “Hills” and “Valleys” can occur in 2 ways
– Smooth hill or valley : derivative is 0 (i.e. horizontal
tangent line at the high or low point)
– Sharp hill or valley: derivative is undefined (i.e. function
is not differentiable at that point)
Where the derivative is 0 or undefined is what is called a critical number.
When referring to both relative max and relative min, we simply say relative extrema

4. Difference between Relative and Absolute
– Relative extrema occur only at critical numbers (Not at the endpoints)
– absolute extrema may occur in the middle or at the endpoints. Must check endpoints if given an interval

5. Absolute maximum
(also local maximum)
Local maximum
Local minimum
Absolute minimum

6. Example
Absolute
Maximum
Absolute Minimum

7. Example
Absolute
Maximum
No Minimum

8. Example No
Maximum
No Minimum

9. Example

10. Example

11. Example

12. Example of some text books including endpoints as local extrema

13. Example

14. Example