Unit 5

1. Unit 5 Higher Derivatives and Applications of Differential Calculus

2. 5.1 Higher Derivatives

4. 5.1 Higher Derivatives

5. 5.1 Higher Derivatives

6. P.156 Ex.5A

7. 5.2 Leibniz’s Rule

8. P.161 Ex.5B

9. 5.3 Maxima and Minima

10. 5.3 Maxima and Minima Any open interval whose centre is x0 and with any length of radius  is a neighbourhood of x0.

11. 5.3 Maxima and Minima

12. 5.3 Maxima and Minima Examples of local maximum :

13. 5.3 Maxima and Minima

14. 5.3 Maxima and Minima Examples of local minimum :

15. 5.3 Maxima and Minima A function f(x) having a relative maximum or minimum at x = x0 need not be differentiable there.

16. 5.3 Maxima and Minima If f(x) is a function differentiable at x0 and if f(x) has a maximum or a minimum at x0, then f ’(x0) = 0.

17. 5.3 Maxima and Minima The converse of the theorem may not be true. There may be points at which f ’(x) vanishes, but f(x) possesses neither a maximum nor a minimum there. An example to verify this fact is f(x) = x3 at x = 0.

18. 5.3 Maxima and Minima When f(x) attains a maximum (resp. minimum) at x = x0, then the value f(x0) is called a maximum value (rep. minimum value) of f(x). Maximum or minimum values are called extreme values.

19. 5.3 Maxima and Minima A point on the graph of y = f(x) at which the function f(x) attains a maximum (resp. minimum) is called a maximum point (resp. minimum point). A turning point is a maximum or minimum point.

20. 5.3 Maxima and Minima The value f(x0) such that f ’(x0) is called a critical value or stationary value and the corresponding point (x0, f(x0)) is called a critical point or stationary point of the graph y = f(x). A tangent at the stationary point of a curve is horizontal. A turning point of the graph of y = f(x), where f(x) is differentiable, is always a stationary point, but a stationary point need not be a turning point.

21. 5.3 Maxima and Minima Examples of stationary point or/and turning point

22. 5.3 Maxima and Minima A function f(x) has an absolute maximum (resp. absolute minimum) or global maximum (resp. global minimum) at x = x0 if and only if f(x0) f(x), for all x Dom(f), where Dom(f) denotes the domain of the function f (resp. f(x0) f(x), for all x Dom(f)). local maximum local maximum and global maximum local minimum

23. 5.3 Maxima and Minima If a function f(x) attains its absolute maximum (resp. absolute minimum) at x = x0, then the value f(x0) is called the absolute maximum value (resp. absolute minimum value) or the global maximum value (resp. global minimum value) or the greatest value (resp. least value) of f(x).

24. 5.3 Maxima and Minima f(x) has relative maximum at x = x1 and a relative minimum at x = x2, but an absolute maximum at x = b and an absolute minimum at x = a.

25. 5.3 Maxima and Minima f(x) has both a relative maximum at x = x1 and an absolute maximum at x = x0, but it has an absolute minimum at x = a and has no relative minimum.

26. 5.3 Maxima and Minima f(x) has neither relative maximum nor minimum, but it has an absolute maximum at x = b and an absolute minimum at x = a.

27. 5.3 Maxima and Minima • Note : • Local or relative extrema are not necessarily the absolute extrema. • Turning point may occur at points where derivatives do not exist, eg. y = |x| has a minimum point at x = 0. • Stationary points are points at which the derivatives are zero, but they may not be turning points. • f ’(x0) = 0 is not sufficient to conclude that there is a relative extremum at x = x0.

28. 5.3 Maxima and Minima

29. 5.4 Increasing and Decreasing Functions

30. 5.4 Increasing and Decreasing Functions A function f(x) is said to be monotonic on an interval I if and only if f(x) is either monotonic increasing or monotonic decreasing on I.

31. 5.4 Increasing and Decreasing Functions

32. 5.4 Increasing and Decreasing Functions A function f(x) is said to be strictly monotonic on an interval I if and only if f(x) is either strictly monotonic increasing or strictly monotonic decreasing on I.

33. 5.4 Increasing and Decreasing Functions Mean Value Theorem

34. 5.4 Increasing and Decreasing Functions

35. 5.4 Increasing and Decreasing Functions

36. 5.4 Increasing and Decreasing Functions

37. 5.4 Increasing and Decreasing Functions

38. 5.4 Increasing and Decreasing Functions Is the function x2 always strictly increasing on R ?

39. P.173 Ex.5C

40. 5.5 Criteria for Determination of Maxima and Minima Three possibilities for determination of maxima and minima :

41. 5.5 Criteria for Determination of Maxima and Minima Three possibilities for determination of maxima and minima : (1) maxima

42. 5.5 Criteria for Determination of Maxima and Minima Three possibilities for determination of maxima and minima :

43. 5.5 Criteria for Determination of Maxima and Minima Three possibilities for determination of maxima and minima : (2) minima

44. 5.5 Criteria for Determination of Maxima and Minima Three possibilities for determination of maxima and minima :

45. 5.5 Criteria for Determination of Maxima and Minima Three possibilities for determination of maxima and minima : Note : As f ’’(x) = 0, we cannot determine whether the extremum is maximum or minimum. It is because the twice-differentiability of a function f(x) at x = x0 gives no information.

46. P.179 Ex.5D

47. 5.6 The Greatest and Least Values of a Function (1) Greatest (resp. least) value is found by comparing the maximum value (resp. minimum value) and the values of the end points

48. 5.6 The Greatest and Least Values of a Function (2) Greatest (resp. least) value is strictly the maximum value (resp. minimum value)

49. 5.6 The Greatest and Least Values of a Function (3) Greatest (resp. least) value is the value of the end point

50. P.182 Ex.5E