1. Extreme values of functions
Section 4.1
Extreme values of functions

2. Absolute (global) extreme values
Definition 1: Absolute Extreme Values:
Let f be a function with domain D. Then f(c) is the
(a) absolute maximum value on D iff for all x in D.
(b) absolute minimum value on D iff for all x in D.

3. Example 1 Find the extreme values and where they occur. 𝑦= cos 𝑥
𝑦= sin 𝑥

4. Example 2 Function Rule Domain, D Absolute Extrema on D 𝑦= 𝑥 2 −∞,∞
[0, 2]
(0, 2]
(0, 2)

5. Theorem 1: the extreme value theorem

6. Local (Relative) Extreme values
Definition 2: Local Extreme Values:
Let c be an interior point of the domain of the function f. Then f(c) is a
(a) local maximum value at c iff for all x in some open interval containing c.
(b) local minimum value at c iff for all x in some open interval containing c.

7. Finding extreme values
If a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if f’ exists at c, then
𝑓 ′ 𝑥 =0

8. Finding extreme values
Definition 3: A point in the interior of the domain of a function f at which f’ = 0 or f’ does not exist is a critical point of f.

9. Example 3
Find the absolute maximum and minimum values of 𝑓 𝑥 = 𝑥 on the interval [-2, 3].

10. Example 4
Use analytical methods to find the extreme values of the function on the interval and where they occur.
𝑓 𝑥 = ln 𝑥+1 , 0≤𝑥≤3

11. Example 5
Find the extreme values of 𝑓 𝑥 = 1 4− 𝑥 2 .

12. Example 6
Find the extreme values of 𝑓 𝑥 = 5−2 𝑥 2 , 𝑥≤1 𝑥+2, 𝑥>1 .

13. Example 7
Find the extreme values of 𝑓 𝑥 =𝑙𝑛 𝑥 1+ 𝑥 2 .