Title CONTINUITY AND DIFFERENTIABILITY.docx.pdf ✅

ChemContent 5. CONTINUITY AND DIFFERENTIABILITY Single Correct Answer Type 1. Let [x] denotes the greatest integer less than or equal to xand f(x) = [x ]. Then, a) f(x) does not exist b) f(x) is continuous at x = 0 c) f(x) is not differentiable at x = 0 d) f ' (0) = 1 2. The value of f(0) so that may be continuous at is (−e x+2 x ) x x = 0 a) log log 1 ( 2 ) b) 0 c) 4 d) − 1 + log log 2 3. Let f(x) be an even function. Then f'(x) a) Is an even function b) Is an odd function c) May be even or odd d) None of these 4. If f(x) = {[cos cos π x], x < 1 |x − 2|, 2 > x≥1 , then f(x) is a) Discontinuous and non-differentiable at x =− 1 and x = 1 b) Continuous and differentiable at x = 0 c) Discontinuous at x = 1/2 d) Continuous but not differentiable at x = 2 5. If f(x) = { , then is |x+2| (x+2) , x≠ − 2 2, x =− 2 f(x) a) Continuous at x =− 2 b) Not continuous x =− 2 c) Differentiable at x =− 2 d) Continuous but not derivable at x =− 2 6. If f(x) = | log log |x| |, then a) f(x) is continuous and differentiable for all x in its domain b) f(x) is continuous for all x in its domain but not differentiable at x = ±1 c) f(x) is neither continuous nor differentiable at x = ±1 d) None of the above 7. If f and , then equals ' (a) = 2 f(a) = 4 xf(a)−af(x) x−a a) 2a − 4 b) 4 − 2a c) 2a + 4 d) None of these 8. If f(x) = x( x + x + 1), then Page| 1
ChemContent a) f(x) is continuous but not differentiable at x = 0 b) f(x) is differentiable at x = 0 c) f(x) is not differentiable at x = 0 d) None of the above 9. If f(x) = {ax , then, is continuous and differentiable at 2 + b, b≠0, x≤1 x 2 b + ax + c, x > 1 f(x) x = 1, if a) c = 0, a = 2b b) a = b, c ∈ R c) a = b , c = 0 d) a = b, c≠0 10. For the function f(x) = {|x − 3|, x≥1 which one of the following is incorrect? x 2 4 − 3 x 2 + 13 4 , x < 1 a) Continuous at x = 1 b) Derivable at x = 1 c) Continuous at x = 3 d) Derivable at x = 3 11. If f: R → R is defined by f(x) = { 2sinsin x−sinsin 2x 2xcoscos x a, if x = 0 , if x≠0, Then the value of a so that f is continuous at 0 is a) 2 b) 1 c) -1 d) 0 12. f(x) = x + |x| is continuous for a) x∈(− ∞, ∞) b) x ∈ (− ∞, ∞) − {0} c) Only x > 0 d) No value of x 13. If the function f(x) = {{1 + |sin sin x |} a |sinsin x | , − π 6 < x < 0 b, x = 0 e tantan 2x tantan 3x , 0 < x < π 6 Is continuous at x = 0 a) a = b , b = 2 3 b) b = a , a = 2 3 c) a = b , b = 2 d) None of these 14. If f(x) = x then at 2 + x 2 1+x 2 + x 2 1+x 2 ( ) 2 + ... + x 2 1+x 2 ( ) n + ..., x = 0, f(x) a) Has no limit b) Is discontinuous c) Is continuous but not differentiable d) Is differentiable 15. Letf(x) = {1, ∀ x < 0 1 + sin sin x, ∀ 0≤x ≤ π/2 , then what is the value of f at ? ' (x) x = 0 a) 1 b) − 1 c) ∞ d) Does not exist 16. The function f(x) = x − |x − x is 2 | a) Continuous at x = 1 b) Discontinuous at x = 1 c) Not defined at x = 1 d) None of the above 17. If f(x + y + z) = f(x). f(y). f(z) for all x, y, z and f(2) = 4, f , then equals ' (0) = 3 f'(2) a) 12 b) 9 c) 16 d) 6 18. If f(x) = ||x| |, then f'(x) equals a) 1 |x| , x≠0 Page| 2
ChemContent b) for and for 1 x |x| > 1 −1 x |x| < 1 c) for and for −1 x |x| > 1 1 x |x| < 1 d) for and for 1 x |x| > 0 − 1 x x < 0 19. If the function f(x) = { is continuous at , then the value of is 1−coscos x x 2 , forx≠0 k, forx = 0 x = 0 k a) 1 b) 0 c) 1 2 d) -1 20. Function f(x) = |x − 1| + |x − 2|, x ∈ R is a) Differentiable everywhere in R b) Except x = 1 and x = 2 differentiable everywhere in R c) Not continuous at x = 1 and x = 2 d) Increasing in R 21. The set of points where the function f(x) = 1 − e is differentiable is −x 2 a) (− ∞, ∞) b) (− ∞, 0)∪(0, ∞) c) (− 1, ∞) d) None of these 22. If f(x) = x sin sin , then the value of function at , so that the function is continuous at 1 ( x ) , x≠0 x = 0 x = 0 is a) 1 b) − 1 c) 0 d) Indeterminate 23. The value of f(0) so that the function f(x) = is continuous everywhere, is given by 2−(256−7 x) 1/8 (5x+32) 1/5−2 (x≠0) a) − 1 b) 1 c) 26 d) None of these 24. The derivative of f(x) = |x| at is 3 x = 0 a) − 1 b) 0 c) Does not exist d) None of these 25. If f(x) = { is continuous function at , then the value 4 x ( −1) 3 sinsin x ( a )loglog 1+ x 2 ( 3 ) , x≠0 9(log log 4 ) 3 , x = 0 x = 0 of a is equal to a) 3 b) 1 c) 2 d) 0 26. f(x) = |[x] + x| in − 1 < x≤2 is a) Continuous at x = 0 b) Discontinuous at x = 1 c) Not differentiable at x = 2, 0 d) All the above 27. Letf(x) = [x , where the greatest integer function is. Then the number of points in the interval (1, 3 − x] [x] 2), where function is discontinuous is a) 4 b) 5 c) 6 d) 7 28. If y = cos(|x| − f(x)) , where Page| 3