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1. Mathematics

2. Session Applications of Derivatives - 2

3. Session Objectives • Increasing and Decreasing Functions • Use of Derivative • Maximum and Minimum • Extreme and Critical points • Theorem 1 and 2 • Greatest and Least Values • Class Exercise

4. Increasing Function

5. Increasing Function A function is said to be a strictly increasing function of x on (a, b). ‘Strictly increasing’ is also referred to as ‘Monotonically increasing’.

6. Decreasing Function

7. Decreasing Function A function ƒ(x) is said to be a strictly decreasing function of x on (a, b). ‘Strictly decreasing’ is also referred to as ‘Monotonically decreasing’.

8. Use of Derivative Let f(x) be a differentiable real function defined on an open interval (a, b).

9. Y T Y = f(x) P q X a O b T' Figure 1 for all x in (a, b). Use of Derivative (Con.) Slope of tangent at any point in (a, b) > 0

10. Y T a P b q X O T' Figure 2 for all x in (a, b). Use of Derivative (Con.) Slope of tangent at any point in (a, b) < 0

11. Example-1 • For the function f(x) = 2x3 – 8x2 + 10x + 5, find the intervals where • f(x) is increasing • (b) f(x) is decreasing

12. Solution We have

13. For x < 1, is positive. is increasing for x < 1 and and it decreases for Solution Cont.

14. Find the intervals in which the function in increases or decreases. Example-2

15. Maximum and Minimum

16. for all small values of If and for all small values of If and Let be a function Maximum and Minimum The point a is called the point of maximum of the function f(x). In the figure, y = f(x) has maximum values at Q and S. The point b is called the point of minimum of the function f(x). In the figure, y = f(x) has minimum values at R and T.

17. At these points, Extreme Points The points of maximum or minimum of a function are called extreme points.

18. The points at which or at whichdoes not exist are called critical points. Critical Points A point of extremum must be one of the critical points, however, there may exist a critical point, which is not a point of extremum.

19. Let the function be continuous in some interval containing x0 . (i) If when x < x0 and When x > x0 then f(x) has maximum value at x = x0 (ii) If when x < x0 and When x > x0 ,then f(x) has minimum value at x = x0 Theorem - 1

20. If x0 be a point in the interval in which y = f(x) is defined and if Theorem - 2

21. Greatest and Least Values The greatest or least value of a continuous function f(x) in an interval [a, b] is attained either at the critical points or at the end points of the interval. So, obtain the values of f(x) at these points and compare them to determine the greatest and the least value in the interval [a, b].

22. Find all the points of maxima and minima and the corresponding maximum and minimum values of the function: (CBSE 1993) Example-3

23. Solution We have For maximum or minimum f’(x) = 0

24. At x = 0, \ f(x) is maximum at x = 0 The maximum value at x = 0 is f(0) = 105 At x = -3, Solution Cont. \ f(x) is minimum at x = -3 The minimum value at x = -3 is

25. At x = -5, \ f(x) is maximum at x = -5 Solution Cont. The maximum value at x = -5 is

26. Example-4 Show that the total surface area of a cuboid with a square base and given volume is minimum, when it is a cube. Solution: Let the cuboid has a square base of edge x and height y.

27. Con.

28. Con.

29. Con.

30. Thank you