1. 4.1 Extreme Values of Function Calculus

2. Extreme Values of a function are created when the function changes from increasing to decreasing or from decreasing to increasing Extreme value decreasin g increasing Extreme value decreasin g   de c inc Extreme value inc de c inc de c Extreme value

3. Absolute Minimum – the smallest function value in the domain Absolute Maximum – the largest function value in the domain Local Minimum – the smallest function value in an open interval in the domain Local Maximum – the largest function value in an open interval in the domain Classifications of Extreme Values Absolute Minimum Absolute Maximum Local Minimum Local Maximum

4. Absolute Minimum at c c Absolute Maximum at c c Definitions: Local Minimum at c c a b Local Maximum at c c a b

5. The number f(c) is called the maximum value of f on D f(c) cd The number f(d) is called the maximum value of f on D f(d) The maximum and minimum values of are called the extreme values of f.

6. Example1:

7. Example2:

8. Example3:

9. The Extreme Value Theorem (Max-Min Existence Theorem) If a function is continuous on a closed interval, [a, b], then the function will contain both an absolute maximum value and an absolute minimum value. abc Absolute maximum value: f(a) Absolute minimum value: f(c)

10. The Extreme Value Theorem (Max-Min Existence Theorem) If a function is continuous on a closed interval, [a, b], then the function will contain both an absolute maximum value and an absolute minimum value. ab d Absolute maximum value: f(c) Absolute minimum value: f(d) c

11. The Extreme Value Theorem (Max-Min Existence Theorem) If a function is continuous on a closed interval, [a, b], then the function will contain both an absolute maximum value and an absolute minimum value. Absolute maximum value: none Absolute minimum value: f(d) ab d c F is not continuous at c. Theorem does not apply.

12. The Extreme Value Theorem (Max-Min Existence Theorem) If a function is continuous on a closed interval, [a, b], then the function will contain both an absolute maximum value and an absolute minimum value. Absolute maximum value: f(c) Absolute minimum value: f(d) F is not continuous at c. Theorem does not apply. ab d c 

13. 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.

14. 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. Finding Maximums and Minimums Analytically: 1Find the derivative of the function, and determine where the derivative is zero or undefined. These are the critical points. 2Find the value of the function at each critical point. 3Find values or slopes for points between the critical points to determine if the critical points are maximums or minimums. 4For closed intervals, check the end points as well.

16. A

17. F081

18. F083

19. F091

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

22. F092

23. F091

24. F081

26. F092

27. F081

28. F083

30. a b c d a27 b0 c0 d-5 a-30 b5 c0 d-7 a-22 b0 c0 d-9 Which table best describes the graph? Table A Table B Table C