1. APPLICATIONS OF DERIVATIVES
CHAPTER: 4
APPLICATIONS OF DERIVATIVES

2. Sec 4.1: Extreme Values of Functions
Outline:
What are global and local maximum, minimum?
Why? The extreme value theorem
How? How to find max and min value?
Fermat’s theorem
Critical points
The closed interval method 2

3. Sec 4.1: Extreme Values of Functions
absolute maximum
global maximum
local maximum
relative maximum
How many local maximum ??

4. Sec 4.1: Extreme Values of Functions
local minimum
relative minimum
absolute minimum
global minimum
How many local minimum ??

5. Sec 4.1: Extreme Values of Functions
The number f(c) is called the maximum value of f on D
f(c) c d
f(d)
The number f(d) is called the minimum value of f on D
The maximum and minimum values of f are called the extreme values of f.

6. Sec 4.1: Extreme Values of Functions
EXAMPLE:
EXAMPLE:

7. Sec 4.1: Extreme Values of Functions
EXAMPLE:
EXAMPLE:

8. Sec 4.1: Extreme Values of Functions
local max at a,b,c
REMARK:
A function f has a local maximum at the endpoint b if
for all x in some half-open interval

9. Sec 4.1: Extreme Values of Functions
local min at a,b,c
REMARK:
A function f has a local minimum at the endpoint a if
for all x in some half-open interval

10. Sec 4.1: Extreme Values of Functions

11. 1 2 Sec 4.1: Extreme Values of Functions
We have seen that some functions have extreme values, whereas others do not. 1
f(x) is continuous on 2
Closed interval [a, b]
attains an absolute maximum f(c) and minimum f(d) value
THE EXTREME VALUE THEOREM

12. 1 2 Sec 4.1: Extreme Values of Functions THE EXTREME VALUE THEOREM
f(x) is continuous on 2
Closed interval [a, b]
attains an absolute maximum f(c) and minimum f(d) value
THE EXTREME VALUE THEOREM
Max?? Min?? What cond??
Max?? Min?? What cond??

13. Sec 4.1: Extreme Values of Functions
f(x) is continuous on 2
Closed interval [a, b]
attains an absolute maximum f(c) and minimum f(d) value
THE EXTREME VALUE THEOREM
Remark:
The Extreme Value Theorem says that a continuous function on a closed interval has a maximum value and a minimum value, but it does not tell us how to find these extreme values. 13

14. Sec 4.1: Extreme Values of Functions
interior point
end point
end point

15. Sec 4.1: Extreme Values of Functions

16. Sec 4.1: Extreme Values of Functions
Remark:
If f has a local maximum or minimum value at an interior point c then c is critical.

17. Sec 4.1: Extreme Values of Functions
The only places where a function ƒ can possibly have an extreme value (local or global) are
interior points where ƒ’= 0
interior points where is ƒ’ undefined,
3. endpoints of the domain of ƒ.

18. Sec 4.1: Extreme Values of Functions
How to find absolute Max and Min
Find all critical points ( ƒ’= 0 , or ƒ’ undefined )
Evaluate ƒ at all criticals
Evaluate ƒ at endpoints a and b
Take the largest value and the smallest

19. Sec 4.1: Extreme Values of Functions
How to find absolute Max and Min
Find all critical points ( ƒ’= 0 , or ƒ’ undefined )
Evaluate ƒ at all criticals
Evaluate ƒ at endpoints a and b
Take the largest value and the smallest

20. Sec 4.1: Extreme Values of Functions
How to find absolute Max and Min
Find all critical points ( ƒ’= 0 , or ƒ’ undefined )
Evaluate ƒ at all criticals
Evaluate ƒ at endpoints a and b
Take the largest value and the smallest

21. Sec 4.1: Extreme Values of Functions
EXAMPLE:
Find:
Local max and min
Global max and min
How to find absolute Max and Min
Find all critical points ( ƒ’= 0 , or ƒ’ undefined )
Evaluate ƒ at all criticals
Evaluate ƒ at endpoints a and b
Take the largest value and the smallest

22. Sec 4.1: Extreme Values of Functions
EXAMPLE:
EXAMPLE:
Find: Absolute max and min
Find: Absolute max and min

23. Sec 4.1: Extreme Values of Functions
How to find absolute Max and Min
(not closed interval)
How to find absolute Max and Min
Find all critical points ( ƒ’= 0 , or ƒ’ undefined )
Evaluate ƒ at all criticals
Evaluate ƒ at endpoints a and b
Take the largest value and the smallest
EXAMPLE:
Find:
Absolute max and min

24. Sec 4.1: Extreme Values of Functions
EXAMPLE:
Find:
Absolute max and min
EXAMPLE:
Find:
Absolute max and min
Remarks:
Polynomial with odd degree  no absolute max, no absoulte min
Polynomial with even degree  absolute max only or absoulte min only

25. How many local maximum How many local minimum
Sec 4.1: Extreme Values of Functions
How many local maximum
How many local minimum

26. Sec 4.1: Extreme Values of Functions
EXAMPLE:
Find:
Absolute max and min
EXAMPLE:
Find:
Absolute max and min
Remarks:
Polynomial with odd degree  no absolute max, no absoulte min
Polynomial with even degree  absolute max only or absoulte min only

27. Sec 4.1: Extreme Values of Functions

28. Sec 4.1: Extreme Values of Functions

29. Sec 4.1: Extreme Values of Functions

30. Sec 4.1: Extreme Values of Functions