Sec4.1: MAXIMUM AND MINIMUM VALUES

1. Sec4.1: MAXIMUM AND MINIMUM VALUES

2. Sec4.1: MAXIMUM AND MINIMUM VALUES absolute maximum global maximum local maximum relative maximum How many local maximum ??

3. Sec4.1: MAXIMUM AND MINIMUM VALUES local minimum relative minimum absolute minimum global minimum How many local minimum ??

4. 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 maximum value of f on D The maximum and minimum values of are called the extreme values of f.

5. c2 c1

6. Example1:

7. Example2:

8. Example3:

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

10. THE EXTREME VALUE THEOREM attains an absolute maximum f(c) and minimum f(d) value 1 f(x) is continuous on 2 Closed interval [a, b] Max?? Min?? What cond?? Max?? Min?? What cond??

11. THE EXTREME VALUE THEOREM attains an absolute maximum f(c) and minimum f(d) value 1 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. f(x) is continuous on 2 Closed interval [a, b]

12. Fermat’s Theorem is named after Pierre Fermat (1601–1665), a French lawyer who took up mathematics as a hobby. Despite his amateur status, Fermat was one of the two inventors of analytic geometry (Descartes was the other). His methods for finding tangents to curves and maximum and minimum values (before the invention of limits and derivatives) made him a forerunner of Newton in the creation of differential calculus.

13. Sec 3.11 HYPERBOLIC FUNCTIONS The following examples caution us against reading too much into Fermat’s Theorem. We can’t expect to locate extreme values simply by setting f’(x) = 0 and solving for x. Exampe5: Exampe6: WARNING Examples 5 and 6 show that we must be careful when using Fermat’s Theorem. Example 5 demonstrates that even when f’(c)=0 there need not be a maximum or minimum at . (In other words, the converse of Fermat’s Theorem is false in general.) Furthermore, there may be an extreme value even when f’(c)=0 does not exist (as in Example 6).

15. F092

16. F081

17. F083

18. F091

19. In terms of critical numbers, Fermat’s Theorem can be rephrased as follows (compare Definition 6 with Theorem 4):

21. F092

22. F091

23. F081

24. F081

25. F092

26. F081

27. F083

28. F083