Chapter 4.4 Increasing and Decreasing Functions PDF
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This document explains strictly increasing and strictly decreasing functions. It provides examples and solutions for determining intervals where functions increase or decrease. The document covers mathematical definitions, illustrating these concepts with graphs and calculus.
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60 Application of Derivatives 219 4.4.1 Definition. E3 (1) Strictly increasing function : A function f(x) is said to be a strictly increasing function on (a, b), if x 1 x 2 f (x 1 ) f (x 2 ) for all x1, x2 (a, b). y y = f(x) Thus, f (x ) is strictly increasing on (a, b), if the values of f (...
60 Application of Derivatives 219 4.4.1 Definition. E3 (1) Strictly increasing function : A function f(x) is said to be a strictly increasing function on (a, b), if x 1 x 2 f (x 1 ) f (x 2 ) for all x1, x2 (a, b). y y = f(x) Thus, f (x ) is strictly increasing on (a, b), if the values of f (x ) ID increase with the increase in the values of x. f(x1) (2) Strictly decreasing function : A function f (x ) is said to be a strictly decreasing function on (a, b) , if x 1 x 2 f (x 1 ) f (x 2 ) for all O a f(x2) x1 x2 b x U x 1 , x 2 (a, b). Thus, f (x ) is strictly decreasing on (a, b), if the values of f (x ) decrease with the increase in the values of x. D YG y f(x1 f(x2 ) ) x O x1 x2 b On the interval (1, 3) the function f (x ) 3 x U Example: 1 a (a) Strictly decreasing x 2 is x [AMU 1999] (b) Strictly increasing ST (c) Decreasing in (2, 3) only (d) Neither decreasing Solution: (b) f (x ) 3 x 2 2 f (x ) 3 2 x x Clearly f (x ) 0 on the interval (1, 3) f (x ) is strictly increasing. Example: 2 Solution: (c) For which value of x, the function f (x ) x 2 2 x is decreasing (a) x 1 (b) x 2 (c) x 1 (d) x 2 y f (x ) (x 1)2 1 Hence decreasing in x 1 O (1, 0) (1, – 1) y= f(x) x increasing nor 220 Application of Derivatives Alternative method: f (x ) 2 x 2 2(x 1) To be decreasing, 2(x 1) 0 ( x 1) 0 x 1. 2 x 3 18 x 2 96 x 45 0 is an increasing function when (a) x 8, x 2 Solution: (a) (b) x 2, x 8 (d) 0 x 2 (c) x 2, x 8 f (x ) 6 x 2 36 x 96 0 , for increasing 60 Example: 3 f (x ) 6(x 8 )(x 2) 0 x 2, x 8. The function x x is increasing, when (a) x 1 e Solution: (a) Let y x x (b) x [MP PET 2003] 1 e (c) x 0 dy dy x x (1 log x ) ; For 0 dx dx 1 e ID x x (1 log x ) 0 1 log x 0 log e x log e 1. e U For this to be positive, x should be greater than 4.4.2 Monotonic Function. (d) For all real x E3 Example: 4 A function f (x ) is said to be monotonic on an interval (a, b) if it is either increasing or D YG decreasing on (a, b). (1) Monotonic increasing function : A function is said to be a monotonic increasing function in defined interval if, y y x 1 x 2 f (x 1 ) f (x 2 ) or x 1 x 2 f (x 1 ) f (x 2 ) x 1 x 2 f (x 1 ) f (x 2 ) or x 1 x 2 f (x 1 ) f (x 2 ) U or O x x O ST (2) Monotonic decreasing function: A function is said to be a monotonic decreasing function in defined interval, if x 1 x 2 f (x 1 ) f (x 2y) y or x1 x 2 f (x1 ) f (x 2 ) or x 1 x 2 f (x 1 ) f (x 2 ) or x 1 x 2 f (x 1 ) f (x 2 ) O Example: 5 The function f (x ) cos x 2 px is monotonically decreasing for x x O [MP PET 2002] Application of Derivatives 221 (a) p (b) p 1 1 sin x p 0 p 2 2 [ 1 sin x 1] If f (x ) x 5 20 x 3 240 x , then f (x ) satisfies which of the following (b) It is monotonically decreasing only in (0, ) (c) It is monotonically increasing every where (d) It is monotonically increasing only in (, 0) f (x ) 5 x 4 60 x 2 240 = 5(x 4 12 x 2 48 ) 5[(x 2 6)2 12] i.e., f (x ) is monotonically increasing everywhere. (b) a 2 (a) a 2 ID The value of a for which the function (a 2)x 3 3ax 2 9 ax 1 decrease monotonically throughout for all real x, are [Kurukshetra CEE 2002] (c) 3 a 0 (d) a 3 If f (x ) (a 2)x 3 3ax 2 9 ax 1 decreases monotonically for all x R , then f ( x ) 0 for all x R U Solution: (d) [Kurukshetra CEE 1996] (a) It is monotonically decreasing everywhere f (x ) 0 x R Example: 7 (d) p 2 60 Solution: (c) (c) p 2 f (x ) will be monotonically decreasing, if f ( x ) 0. f (x ) sin x 2 p 0 Example: 6 1 2 E3 Solution: (b) 1 2 3(a 2)x 2 6 ax 9 a 0 for all x R (a 2)x 2 2ax 3 a 0 for all x R a 2 0 and discriminant 0 a 2 and 8 a 2 24 a 0 Example: 8 D YG a 2 and a(a 3) 0 a 2 and a 3 or a 0 a 3 a 3 Function f (x ) sin x 6 cos x 2 sin x 3 cos x (a) 1 Solution: (d) is monotonic increasing if (c) 4 (b) 1 (d) 4 The function is monotonic increasing if, f ( x ) 0 (2 sin x 3 cos x )( cos x 6 sin x ) ( sin x 6 cos x ) (2 cos x 3 sin x ) – 0 (2 sin x 3 cos x )2 (2 sin x 3 cos x )2 U 3(sin 2 x cos 2 x ) 12(sin 2 x cos 2 x ) 0 3 12 0 4. ST 4.4.3 Necessary and Sufficient Condition for Monotonic Function. In this section we intend to see how we can use derivative of a function to determine where it is increasing and where it is decreasing (1) Necessary condition : From figure we observe that if f (x ) is an increasing function on (a, b), then tangent at every point on the curve y f (x ) makes an acute angle with the y y P(x, y) positive direction of x-axis. tan 0 dy 0 or f ' (x ) 0 dx P(x,y) for all x (a, b) It is evident from figure that if f (x ) is a y=f(x) y=f(x) O a b x O a x b 222 Application of Derivatives decreasing function on (a, b), then tangent at every point on the curve y = f(x) makes an obtuse angle with the positive direction of x-axis. tan 0 dy 0 or f ' (x ) 0 for all x (a, b). dx 60 Thus, f ' (x ) 0( 0) for all x (a, b) is the necessary condition for a function f (x) to be increasing (decreasing) on a given interval (a, b). In other words, if it is given that f (x) is increasing (decreasing) on (a, b), then we can say that f ' (x ) 0 ( 0 ). E3 (2) Sufficient condition : Theorem : Let f be a differentiable real function defined on an open interval (a, b). (a) If f ' ( x ) 0 for all x (a, b) , then f (x ) is increasing on (a, b). (b) If f ' (x ) 0 for all x (a, b) , then f (x ) is decreasing on (a, b). Corollary : Let f (x) be a function defined on (a, b). ID (a) If f ' (x ) 0 for all x (a, b) , except for a finite number of points, where f ' (x ) 0, then f (x ) is increasing on (a, b). decreasing on (a, b). The function f (x ) ln( x ) is ln(e x ) D YG Example: 9 U (b) If f ' (x ) 0 for all x (a, b) , except for a finite number of points, where f ' (x ) 0, then f (x) is (a) Increasing on [0, ) (b) Decreasing on [0, ) (c) Decreasing on 0, and increasing on , e e (d) Increasing on 0, and e decreasing on , e Let f (x ) ln( x ) ln(e x ) 1 1 ln( x ) x e x = (e x ) ln(e x ) ( x ) ln( x ) {ln(e x )}2 {ln(e x )}2 (e x )( x ) U Solution: (b) f (x ) ln(e x ) ST f ( x ) 0 for all x 0 Example: 10 Which of the following is not a decreasing function on the interval 0, 2 (a) cos x Solution: (c) { e}. Hence, f (x ) is decreasing in [0, ). (b) cos 2 x (c) cos 3 x (d) cot x d cos 3 x 3 sin 3 x. Obviously, here cos 3 x in not decreasing in 0, because 2 dx But at x 75 o , 3 sin 3 x 0. Hence the result. Example: 11 2 The interval of increase of the function f (x ) x e x tan is 7 (a) (0, ) (b) (, 0) (c) (1, ) (d) (, 1) Application of Derivatives 223 Solution: (b, d) 2 We have f (x ) x e x tan f (x ) 1 e x 7 For f (x ) to be increasing, we must have f ( x ) 0 1 e x 0 e x 1 x 0 x (, 0) (, 1) (, 0) 60 4.4.4 Test for Monotonicity. (1) At a point : (i) Function f (x) will be monotonic increasing in domain at a point if and only if, f ' (a) 0 E3 (ii) Function f (x) will be monotonic decreasing in domain at a point if and only if, f ' (a) 0. (2) In an interval : Function f (x), defined in [a, b] is f ' (x ) 0 , (i) Monotonic increasing in (a, b) if, ax b Monotonic increasing in [a, b] if, f ' (x ) 0 , a x b (iii) Strictly increasing in [a, b], if, f ' (x ) 0 , a x b (iv) Monotonic decreasing in (a, b), if, f ' (x ) 0 , ax b f ' (x ) 0 , Strictly decreasing in [a, b], if, f (x ) xe x (1 x ) then f (x ) is D YG Example: 12 1 (a) Increasing on , 1 (b) Decreasing on R 2 Solution: (a) ax b f ' (x ) 0 , U (v) Monotonic decreasing in [a, b], if, (vi) ID (ii) ax b (c) Increasing on R [IIT Screening 2001] 1 (d) Decreasing on , 1 2 f (x ) e x (1 x ) x. e x (1 x ). (1 2 x ) = e x (1 x ){1 x (1 2 x )} e x (1 x ). (2 x 2 x 1) Now by the sign-scheme for 2 x 2 x 1 + U – – 1/2 – 1 ST 1 f (x ) 0, if x , 1 , because e x (1 x ) is always positive. So, f (x ) is increasing on 2 Example: 13 x tends 0 to then the given function f (x ) x sin x cos x cos 2 x is (a) Increasing (b) Decreasing (c) Neither increasing nor decreasing Solution: (b) (d) None of these f (x ) x sin x cos x cos x 2 f (x ) sin x x cos x sin x 2 cos x sin x = cos x (x 2 sin x ) Hence x 0 to , then f ( x ) 0 , i.e., f (x ) is decreasing function. 4.4.5 Properties of Monotonic Function. 1 2 ,1. 224 Application of Derivatives (1) If f (x ) is strictly increasing function on an interval [a, b], then f 1 exists and it is also a strictly increasing function. (2) If f (x ) is strictly increasing function on an interval [a, b] such that it is continuous, then f 1 is continuous on [ f (a), f (b )] 60 (3) If f (x ) is continuous on [a, b] such that f (c) 0( f (c) 0) for each c (a, b ), then f (x ) is monotonically (strictly) increasing function on [a, b]. monotonically (strictly) decreasing function on [a, b] E3 (4) If f (x ) is continuous on [a, b] such that f (c) 0( f (c) 0) for each c (a, b ) , then f (x ) is (5) If f (x ) and g(x ) are monotonically (or strictly) increasing (or decreasing) functions on [a, b], then gof(x) is a monotonically (or strictly) increasing function on [a, b] ID (6) If one of the two functions f (x ) and g(x ) is strictly (or monotonically) increasing and other a strictly (monotonically) decreasing, then gof(x) is strictly (monotonically) decreasing on [a, b]. The interval in which the function x 2 e x is non decreasing, is U Example: 14 (a) (, 2] Let f (x ) x 2 e x D YG Solution: (b) (c) [2, ) (b) [0, 2] (d) None of these dy 2 xe x x 2 e x e x (2 x x 2 ) dx Hence f ( x ) 0 for every x [0, 2] , therefore it is non-decreasing in [0, 2]. Example: 15 The function sin 4 x cos 4 x increase if (b) 8 4 x 3 8 U (a) 0 x f (x ) sin 4 x cos 4 x = (sin 2 x cos 2 x )2 2 sin 2 x cos 2 x = 1 4 sin 2 x cos 2 x sin 2 2 x 1 1 1 (2 sin 2 2 x ) 2 2 4 ST Solution: (b) 1 cos 4 x 3 1 = 1 cos 4 x 4 4 4 Hence function f (x ) is increasing when f ( x ) 0 f (x ) sin 4 x 0 sin 4 x 0 Hence 4 x 3 3 or. x 2 4 8 [IIT 1999] (c) 3 5 x 8 8 (d) 5 3 x 8 4 U D YG U ID E3 60 Application of Derivatives 225 The function x (a) [– 1, 1] 2. If f (x ) sin x (a) 0 x 4. (b) [0, 1] (c) [– 1, 0] (d) [–1, 2] The interval for which the given function f (x ) 2 x 3 3 x 2 36 x 7 is decreasing, is (a) (– 2, 3) 3. Basic Level 1 ( x 0 ) is a non-increasing function in the interval x ST 1. Increasing and Decreasing Function 3 (b) (2, 3) (c) (2, –3) (d) None of these x is increasing function, then 2 (b) 3 x 0 [MP PET 1987] (c) 3 x 3 (d) x 2 If the function f : R R be defined by f (x ) tan x x , then f (x ) (a) Increases (b) Decreases (c) Remains constant (d) Becomes zero 226 Application of Derivatives 2 x 3 6 x 5 is an increasing function if (b) 1 x 1 (a) 0 x 1 9. [Kerala (Engg.) 2002] (c) x 1 (b) x 1 (c) (, 1) (b) (2, 1) The function f (x ) tan x x (b) Always decreases (c) Never decreases (d) Sometimes increases and sometimes decreases If f (x ) kx 3 9 x 2 9 x 3 is monotonically increasing in each interval, then (b) k 3 (c) – 1 The function f (x ) x cos x is (d) None of these (a) (1,1) U (d) None of these is increasing in the interval (b) (, ) Function f (x ) x 4 (d) – 2 (b) Always decreasing D YG The function f (x ) x 2 ID (b) – 3 (c) Increasing for certain range of x (c) (0, ) (d) (, 0) x3 is 3 (a) Increasing for x 1 1 and decreasing for x 4 4 (c) Decreasing for every value of x (b) Increasing for every value of x (d) None of these The function y 2 x 3 9 x 2 12 x 6 is monotonic decreasing when U 14. [Rajasthan PET 1992; The least value of k for which the function x kx 1 is an increasing function in the interval 1 x 2 is (a) Always increasing 13. (c) k 3 2 (a) – 4 12. (d) (, 2) and (1, ) (a) Always increases (a) k 3 11. [MP PET 1993] [MNR 1995] Kurukshetra CEE 2002] 10. (d) All values of x For which interval, the given function f (x ) 2 x 3 9 x 2 12 x 1 is decreasing (a) (2, ) 8. (d) 1 x 1 / 2 The function f (x ) 1 x 3 x 5 is decreasing for (a) 1 x 5 7. (c) x 1 or x 1 60 6. [UPSEAT 2003] E3 5. [MP PET 1994; Rajasthan PET 1996] (b) x 2 ST (a) 1 x 2 15. The function (c) [2, ) (d) None of these (b) (, 0] (c) [1, ) (d) (0, ) The function sin x bx c will be increasing in the interval (, ) if (a) b 1 18. (b) [0, 2] 1 is decreasing in the interval 1 x2 (a) (, 1] 17. (d) None of these The interval in which the x 2 e x is non-decreasing, is (a) (, 2] 16. (c) x 1 (b) b 0 (c) b 1 In the interval [0, 1], the function x 2 x 1 is (a) Increasing (b) Decreasing (d) b 0 Application of Derivatives 227 (c) Neither increasing nor decreasing f (x ) x 3 27 x 5 is an increasing function, when (b) Increasing (c) Neither increasing nor decreasing (d) Increasing for x 0 and decreasing for x 0 In which interval is the given function f (x ) 2 x 3 15 x 2 36 x 1 is monotonically decreasing (b) (2, 3) Let f (x ) x 3 bx 2 cx d , 0 b 2 c. Then f (b) Has a local maxima (c) Has a local minima (d) (3, ) 1 (d) 1, 3 [IIT JEE Screening 2004] (d) Is strictly increasing The function f (x ) x 3 3 x 2 24 x 5 is an increasing function in the interval given below (a) (, 2) (4, ) (b) (2, ) (c) (2, 4 ) (d) (, 4) Which one is the correct statement about the function f (x ) sin 2 x D YG 25. 1 1 (c) , 3 3 1 (b) , 1 3 (a) Is bounded 24. (c) (, 2) The interval of the decreasing function f (x ) x 3 x 2 x 4 is 1 (a) , 1 3 23. 1 is 5x (a) Decreasing (a) [2, 3] 22. (d) | x | 3 E3 21. For the every value of x the function f (x ) ID 20. (c) x 3 (b) | x | 3 60 (a) x 3 [MP PET 1995] U 19. (d) None of these (a) f (x ) is increasing in 0, and decreasing in , 2 2 (b) f (x ) is decreasing in 0, and increasing in , 2 2 (c) f (x ) is increasing in 0, and decreasing in , 4 2 4 26. U (d) The statement (a), (b) and (c) are all correct If f (x ) x 3 10 x 2 200 x 10 , then [Kurukshetra CEE 1998] ST (a) f (x ) is decreasing in ] , 10 ] and increasing in [10 , [ (b) f (x ) is increasing in ] , 10 ] and decreasing in [10 , [ (c) f (x ) is increasing throughout real line 27. If f is a strictly increasing function, then lim (a) 0 28. f (x 2 ) f (x ) is equal to f (x ) f (0 ) (b) 1 (c) – 1 Function x 6 x 9 x 1 is monotonic decreasing when 3 2 (a) 1 x 3 29. x 0 (d) f (x ) is decreasing throughout real line The function f (x ) (a) x 3 (b) x 3 (d) 2 [Rajasthan PET 1991] (c) x 1 (d) x 3 or x 1 1 3 1 2 x x 6 x 8 is decreasing in the interval 3 2 (b) x 2 (c) 3 x 2 (d) None of these 228 Application of Derivatives The function f (x ) 2 log( x 2) x 2 4 x 1 increases in the interval (a) (1, 2) The function f (x ) Monotonically increasing (c) Constant function In the following decreasing function is (a) ln x 33. (d) (2, 4) | x| (x 0), x 0 is x (a) Monotonically decreasing (b) 32. (c) (, 1) (b) 1 | x| (c) e 1 / x If f (x ) kx sin x is monotonically increasing, then (b) k 1 (a) k 1 (c) k 1 34. The function f defined by f (x ) (x 2)e x is D YG (b) a 1 (b) (5, 1) (b) (2, 1) (c) (1, 5) (d) (5, ) (c) (1, 2) (d) (2, ) U If f (x ) 2 x cot 1 x log( 1 x 2 x ) , then f (x ) (b) Decreases in [0, ) (c) Neither increases nor decreases in (0, ) (d) Increases in (, ) ST (a) Increases in [0, ) The function (e 2 x 1) is (e 2 x 1) (a) Increasing 40. The function (a) ad bc 0 41. (d) a 2 Let f (x ) e x (x 1)(x 2)dx. Then f decreases in the interval (a) (, 2) 39. (c) a 2 The interval in which the function x 3 increases less rapidly then 6 x 2 15 x 5 , is (a) (, 1) 38. U The value of a in order that f (x ) 3 sinx cos x 2ax b decreases for all real values of x, is given by (a) a 1 37. [IIT Screening 1994] (d) Decreasing in (1, ) and increasing in (, 1) (c) Increasing for all x 36. (d) k 1 (b) Decreasing in (, 1) and increasing in (1, ) (a) Decreasing for all x 35. (d) None of these ID Advance Level (d) 60 31. (b) (2, 3) E3 30. [Roorkee 1998] (b) Decreasing (c) Even a sin x b cos x is decreasing if c sin x d cos x (b) ad bc 0 [Rajasthan PET 1999] (c) ab cd 0 If f (x ) sin x cos x , 0 x 2 the function decreasing in 5 3 (a) , 6 4 (b) , 4 2 (d) Odd (d) ab cd 0 [UPSEAT 2001] 3 5 , (c) 2 2 (d) None of these Application of Derivatives 229 43. If f (x ) 1 log(1 x ), x 0 then f is x 1 [Rajasthan PET 2002] (a) An increasing function (b) A decreasing function (c) Both increasing and decreasing function (d) None of these The function f (x ) x 1 / x is [AMU 2002] (a) Increasing in (1, ) (b) Decreasing in (1, ) (c) Increasing in (1, e ) , decreasing in (e, ) (d) Decreasing in (1, e ) increasing in (e , ) 60 42. 44. The length of the longest interval, in which the function 3 sin x 4 sin 3 x is increasing, is (b) The function f (x ) 1 e x 2 /2 2 (c) is 3 2 (d) (a) Decreasing for all x (b) Increasing for all x (c) Decreasing for x 0 and increasing for x 0 (d) Increasing for x 0 and decreasing for x 0 ID 45. 3 E3 (a) 46. The function sin x cos x is increasing in the interval (d) None of these On the interval 0, , the function log sin x is 2 D YG 47. 3 (c) , 4 4 U 3 (b) 0, 4 3 7 , (a) 4 4 (a) Increasing (b) Decreasing (c) Neither increasing nor decreasing (d) None of these 48. For all real values of x, increasing function f (x ) is (a) x 1 (b) x 2 [MP PET 1996] (c) x 3 (d) x 4 3 is 2 2 49. The function which is neither decreasing nor increasing in , 50. (b) tan x U (a) cosec x (d) | x 1| For every value of x, function f (x ) e x is ST (a) Decreasing (c) Neither increasing nor decreasing 51. (c) x 2 (b) Increasing (d) None of these Consider the following statements S and R S : Both sin x and cos x are decreasing functions in , 2 R : If a differentiable function decreases in (a, b) then its derivative also decrease in (a, b) Which of the following is true (a) Both S and R are wrong (b) Both S and R are correct but R is not the correct explanation for S (c) S is correct and R is the correct explanation for S (d) S is correct and R is wrong 230 Application of Derivatives 53. If f (x ) is zero in the interval (a, b) then in this interval it is (a) Increasing function (b) Decreasing function (c) Only for a 0 and b 0 is increasing function (d) None of these The function x 2 , (x 1) is increasing on the interval x 1 (b) [0, ) (a) (, 0] 54. (c) R If f and g are two decreasing functions such that fog exists, then fog (a) Is an increasing function (b) The function f (x ) cos( / x ) is increasing in the interval 1 , 2n , n N (b) 2n 1 (a) (2n 1, 2n), n N 3 21 (b) 4 , (1, ) 2 (a) (, ) 57. The function f (x ) x ax x 2 , a 0 D YG 3a (a) Increases on the interval 0 , 4 3a (c) Decreases on the interval 0 , 4 58. The function f (x ) (a) (2, ) | x 1| x2 27 (2, ) (c) 3, 5 2 (d) [1, ) 3a (b) Decreases on the interval , a 4 3a (d) Increases on the interval , a 4 is monotonically decreasing on (b) (0, 1) (c) (0, 1) (2, ) (d) (, ) The set of values of a for which the function f (x ) x 2 ax 1 is an increasing function on [1, 2] is U 59. (d) None of these a4 1 x 5 3 x log 5 decreases for all real x is The set of all values of a for which the function f (x ) 1a ID 56. 1 1 , (c) , nN 2n 2 2n 1 U 55. (d) None of these Is a decreasing function E3 (c) Is neither increasing nor decreasing (d) None of these 60 52. (b) [4, ] (a) (2, ) (c) [, 2) (d) (, 2] ST 60. On which of the following intervals is the function x 100 sin x 1 decreasing (a) 0, 2 61. (d) None of these (b) x 0 (c) x 1 (d) x 1 y [ x (x 3)]2 increases for all values of x lying in the interval (a) 0 x 63. (c) , 2 If a 0 the function f (x ) e ax e ax is a monotonically decreasing function for values of x given by (a) x 0 62. (b) (0, 1) 3 2 The function f (x ) (b) 0 x log x is increasing in the interval x (c) x 0 (d) 1 x 3 [EAMCET 1994] Application of Derivatives 231 (a) (1, 2e) (b) (0, e ) 1 (d) , 2 e e (c) (2, 2e ) 64. The value of a for which the function f (x ) sin x cos x ax b decreases for all real values of x, is given by (b) a 1 (a) a 2 (a) a b (b) a b 2 (c) a x x and g(x ) , where 0 x 1 , then in this interval tan x sin x 66. If f (x ) (a) Both f (x ) and g(x ) are increasing functions (d) a 3 2 (b) Both f (x ) and g(x ) are decreasing function (c) f (x ) is an increasing function (d) function Let h(x ) f (x ) ( f (x ))2 ( f (x ))3 for every real number x, then g(x ) is an increasing ID 67. 1 2 60 If the function f (x ) cos | x | 2ax b increases along the entire number scale, the range of values of a is given by E3 65. (d) a 1 (c) a 2 (a) h is increasing whenever f is increasing and decreasing whenever f is decreasing (b) h is increasing whenever f is decreasing (d) Nothing can be said in general U (c) h is decreasing whenever f is increasing D YG 2 3 x 12 x 1 , 1 x 2 then f (x ) is 37 x , 2x3 68. If f (x ) (a) Increasing in [–1, 2] 69. (b) Continuous in [–1, 3] (c) Greatest at x 2 [IIT 1993] (d) All of these If f (x ) g(x )(x )2 where g( ) 0 and g (x ) is continuous at x then function f (x ) (a) Increasing near to if g( ) 0 g( ) > 0 (c) Increasing near to if g( ) 0 Decreasing near to (d) Increasing near to for U every value of g( ) Function cos 2 x cos 2 x cos x cos x for all real values of x will be 3 3 ST 70. (a) Increasing 71. If f (x ) (d) None of these 1 (b) 0, 2 1 (c) , 1 2 (d) (0, 1) x c for 5 x 5 , then f (x ) is increasing function in the interval c x (a) [c, 5] 73. (c) Decreasing Let Q(x ) f (x ) f (1 x ) and f (x ) 0 whereas 0 x 1 then function Q(x ) is decreasing in 1 (a) , 1 2 72. (b) Constant (b) [0, c] (c) [c, 0] If the domain of f (x ) sin x is D {x : 0 x } , then f (x ) is (a) Increasing in D (b) (b) Decreasing in D (d) [c, c] if 232 Application of Derivatives (c) Decreasing in 0, and increasing in , 2 2 74. (d) None of these x If f (x ) (ab b 2 1)x (cos 4 sin 4 )d is a decreasing function of x for all x R and b R , b being independent 0 of x, then 75. If f (x ) (c) a ( 6 ,0) (b) a ( 6 , 6 ) p2 1 3 x 3 x log 2 is a decreasing function of x in R then the set of possible values of p (independent of p2 1 (c) (,1] (b) [1, ) (a) [-1, 1] (d) None of these Let f f (x ) a5 x 5 a 4 x 4 a 3 x 3 a 2 x 2 a1 x , where ai' s are real and f ( x ) 0 has a positive root 0. Then (b) f ( x ) 0 has at least two real root (c) f ( x ) 0 has at least one real roots (d) None of these x a2 If a, b, c are real, then f (x ) ab ac ac is decreasing in bc x c2 a2 b 2 c2 , 0 (c) 3 2 (b) 0, (a 2 b 2 c 2 ) 3 D YG 2 (a) (a 2 b 2 c 2 ), 0 3 ab x b2 bc ID (a) f ( x ) 0 has a root 1 such that 0 1 0 U 77. E3 x) is 76. (d) None of these 60 (a) a (0, 6 ) ST U **** (d) None of these E3 Assignment (Basic and Advance Level) 60 232 Application of Derivatives 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 a c a c d d a c d a c a a b d c d b a 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 b b d a c c c a c b c c a d b c c a,d a,d b 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 d b c a c b a c a b d d b a d b a,b c a d 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 b a b a c c a d a b a a d b a a,b, c a ST U D YG U ID 1 a
