1. MAT 3749 Introduction to Analysis
Section 2.3 Part III
The Mean Value Theorem

2. Important Result a b

3. Preview Extreme Value Theorem Fermat’s Theorem Rolle’s Theorem
The Mean Value Theorem

4. References
Section 2.3

5. Maximum Value

6. Local Maximum

7. T or F
An absolute max is a local max.

8. The Extreme Value Theorem

9. Fermat’s Theorem

10. Lemma (HW)

11. Fermat’s Theorem

12. Conceptual Diagrams

13. Fermat’s Theorem

14. Fermat’s Theorem

15. Fermat’s Theorem

16. Fermat’s Theorem

17. Proof

18. Proof

19. Proof

20. Proof

21. Rolle’s Theorem

22. Rolle’s Theorem

23. Rolle’s Theorem

24. Rolle’s Theorem

25. Rolle’s Theorem

26. Rolle’s Theorem

27. Rolle’s Theorem

28. Rolle’s Theorem

29. Rolle’s Theorem

30. Rolle’s Theorem

31. Rolle’s Theorem

32. Rolle’s Theorem

33. Rolle’s Theorem

34. Rolle’s Theorem

35. Rolle’s Theorem

36. Rolle’s Theorem

37. Rolle’s Theorem

38. Rolle’s Theorem

39. Rolle’s Theorem

40. Rolle’s Theorem

41. Proof

42. Proof

43. The Mean Value Theorem

44. Proof

45. The Mean Value Theorem

46. The Mean Value Theorem

47. The Mean Value Theorem

48. Theorem (Consequence)
If f’(x)=0 for all x in an interval (a,b), then f is constant on (a,b).
Q: Can we apply the MVT directly?

49. Corollary (Important)b

50. Corollary (Important)b