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Cengage Daily Practice Problem Class 12 Continuity and Differentiability

Cengage Daily Practice Problem Class 12 Continuity and Differentiability (https://doubtnut.com/cengage-solutions/chapter-continuity-and-differentiability-1) PDfs provides accurate solutions to practice before your final exam. Students can study this comprehensive Cengage Maths PDf Solutions on Continuity and Differentiability and its various sub-topics for effective exam preparation. The Cengage Class 12 PDF solutions help the student to swiftly practice and revise the important concepts just before the annual exam to score high marks in maths.<br>For more such video solutions/Pdf solution, visit https://doubtnut.com and clear your doubt instantly.<br>

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Cengage Daily Practice Problem Class 12 Continuity and Differentiability

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  1. CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER CONTINUITY AND DIFFERENTIABILITY || DPP DAILY PRACTICE PROBLEMS  Download Doubtnut Today Ques No. Question CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous f is a continous function in defined as h(x) = f(x)f or x ∈ [a,b), ; g is a continuous function in [b,c]. A function h(x) is [a,b] 1 g(x)f or x ∈ (b, c] if f(b) =g(b) then  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If the function f(x) defined as defined as f(x) f(x) ⎧ ⎨ ⎩3,x = 0(1 + 1 ax + bx3 x 2 ) , = x2 x > 0 is continuous at then b. c. d. b = e3 x = 0, a = 0 a = 1 b = (log)e3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous A twice differentiable function f(x)is defined for all real numbers and satisfies the following conditions f(0) = 2; f'(0) − − 5 and f(0) 3 = 3 . The function constant If is defined by g(x) , where 'a' is any g(x) = eax+ f(x)∀x ∈ R . Find the value(s) of 'a' g'(0) + g(0) = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If and it follows the relation y = y(x) , then find (i) and (ii) exy+ ycosx = 2 y'(0) (0)

  2. 4 . y(0)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If y = e− xcosxandyn+ kny = 0, dny 5 whereyn= andkn dxn ∀n ∈ N, are constants then (b) (d) k4= 4 k8= − 16 k12= 20 k16= − 24  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If y3− y = 2x, d2y 1 6 dy then(x2− ) + x = dx2 27 d y y y b. c. d. y 3 9 27  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Given f(x) 2x3− 3 = {3 − [cot−1( )]f x2 1 or x > 0{x2}cos(e )f or x x 7 < 0

  3. (where {} and [] denotes the fractional part and the integral part functions respectively). Then which of the following statements do/does not hold good? b. c. if irremovable discontinuity of at f x = 0 , then is continuous at f(x) d. f(0−) = 0 f(0+) = 3 f(0) = 0 x = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If f(x) = {s ∈ (cos− 1x) + cos(sin−1x), x ≤ 0s ∈ (cos−1x) 8 − cos(sin−1x,x > 0) . Then at differentiable differentiable is continuous and differentiable not continuous but differentiable f(x) is continuous but not is neither continuous nor x = 0 f(x) f(x) f(x)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let and g(x) = {sinx + cosx,x < 01, x (where [.] denotes the greatest integer function) f(x) = {[x]x ∈ Ix − 1x ∈ I 9 ≥ 0 . Then for exists but not continuous Continuous f(g(x))atx = 0 ( lim )x→ x = 0 0f(g(x)) but not differentiable at Differentiable at does not x = 0 ( lim )x→ 0f(g(x)) exist is continuous but not differentiable f(x)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let continuous and differentiable for all numbers in its domain then and If is g(x) = 3x2− 4√x + 1,x < 1 g(x) = ax + b,x ≥ 1. g(x) 10  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous f(h) − f(0) If is an even function such that f has some finite non-zero ( lim )h→ 11 0 h value, then prove that is not differentiable at f(x) x = 0.

  4.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If f(x) = {[x] + √{x},x 1 12 < 1 , x ≥ 1 [x] + {x}2 , then [where [.] and {.] represent the greatest integer and fractional part functions respectively] is continuous at differentiable at does not exist x = 1 ( lim )x→ is not continuous at is f(x) x = 1 f(x) x = 1 f(x) 1f(x)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Number of points where f(x) = x2−∣∣x2− 1∣∣+ 2||x| − 1| 13 + 2|x| − 7 is non-differentiable is a. 0 b. 1 c. 2 d. 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let f′(x) < 0on( − 4,6), f′(x) > 0on(6,∞), g′(2) be differentiable for real such that x f′(x) > 0on( − ∞, − 4), f(x) 14 If then the g(x) = f(10 − 2x), value of is a. 1 b. 2 c. 0 d. 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND

  5. DIFFERENTIABILITY_Miscellaneous 4sin2x + 3cos2x The differential coefficient of with respect to sin−1( ) 15 5 5cosx − 4sinx is b. c. d. −1 cos−1( ) −2 1 2 √41  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let such that is a twice differentiable function on The value of equals __________ ∣∣g− π∣∣ g(x) = f(x)sinx,wheref(x) ( − ∞,∞) 16 f( − π) = 1.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous 1 17 If at is equal to (b) e (c) 1 (d) zero f(x) = (log)x(lnx),thenf′(x) x = e e  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If f(x − y) = f(x). g(y) − f(y) .g(x)

  6. 18 and g(x − y) = g(x).g(y) + f(x) .f(y) for all at x =0 . If right handed derivative at x=0 exists for f(x) find the derivative of g(x) x ∈ R  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous dy 19 If then is a. 0 b. 1 c. -1 d. none of these atx = 0 xexy− y = sin2x dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous The right hand derivative of greatest integer function) b. is (where [.] denotes the f(x) = [x]tanπxatx = 7 7π −7π 20 c. d. none of these 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If x2 1 x√x2+ 1 y = + 2 2 21 + (log)e√x +√x2+ 1 2y = xy′+ (log)ey′,wherey' , prove that denotes the derivative w.r.t x.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let the derivatives of theirpair wise products at (fg)'(0) = 6;(gh)'(0) are differentiable functions. If f,g and h and f(0) = 1; g(0) = 2;h(0) = 3 x = 0 are 22 = 4 and (hf)'(0) = 5 then compute the value of . (fgh)'(0)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If for a continuous function 23

  7. f, f(0) = f(1) = 0, f′(1) = 2andy(x) = f(ex)ef (x) y′(0) , then is equal to a. 1 b. 2 c. 0 d. none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If y = xlogx^ ((log(log .x))), dy then is 24 dx y y (logx)log(logx)(2log(logx) + 1) (1nx∞x−1) + 21nx1n(1nx)) x x logy y y [(1nx)2+ 21n(1nx)] [2log(logx) + 1] logx x1nx x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous d [cos−1(x√x − √(1 − x)(1 − x2))] = dx 25 1 1 −1 1 1 1 − − + √1 − x2 1 √1 − x20 2√x − x2 √1 − x2 2√x − x2 √1 − x2 2√x − x2 b. c. d. none of these −1/4 1/4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let

  8. g(x) = ef (x)andf(x + 1) = x + f(x)∀x ∈ R. If 1 g′(n + ) 2 n ∈ I+, then 26 1 g(n + ) 2 1 g′( ) 2 − = 1 g( ) 2 1 1 1 1 1 1 1 ) n 2(1 + ) 2(1 + + + + + + 2 3 3 5 2n − 1 n  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Suppose that Given that G(x) = x2g(x) − xh(g(x))∀x is differentiable invertible function f(1) = f′(1) = 1, h(1) = 0 f′(x) ≠ 0andh′(x) = f(x). f(x) f(x) and is inverse of g(x) . Let 27 ∈ R. Which of the following is/are correct? G1= 3 b. c. d. G1= 2 G′(1) = 2 G′(1) = 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If √1 + x2+ 1 y = cos−1√ , 2√1 + x2 dy isequa < o then 28 dx 1 1 −1 (b) (d) ,x ∈ R , x > 0 ,x < 0 2(1 + x2) 1 2(1 + x2) 2(1 + x2) ,x < 0 2(1 + x2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let 1 f(x) = x[ ] + x[x] if x ≠ 0; x 29 0 if x = 0 where[x] denotes the greatest integer function. then the correct statements are (A) Limit exists for x=-1 (B) f(x) has removable discontonuity at x =1 (C) f(x) has non removable discontinuity at x =2 (D) f(x) is discontinuous at all positive integers

  9.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous dy dx Let If and at then at y = x3− 8x + 7andx = f(t). = 2 x = 3 t = 0, 30 dt dt 19 2 is given by 1 (b) (c) (d) none of these t = 0 2 19  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous 1 is a strictly monotonic differentiable function with If is the g f′(x) = f . √1 + x3 2g2(x) 2x2 x2 31 3 inverse of then a. b. c. d. g2(x) gx= f, 2 2√1 + x3 √1 + x3 2√1 + g2(x)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous + f:R→ Suppose 3f(x + y) = f(x)f(y)∀x, y ∈ R 12 15 be a differentiable with f(1) = 6. function such f(2) that R 32 Then the value of is b. 6 9 c. d.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous In a question a student was given to find the derivative of the product of two functions ′ 3

  10. The student by mistake thought and he got the correct answer. Given that g(5) =1 8 f′(x) < 0 for his question Then which of the following is (fg)′= f'g' f(x) = x3 fandg. 33 g(4) = 1. false? b. c. d. none of these f(0) < 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous . ′(x). A nonzero polynomial with real coefficient has the property that If is the leading coefficient of a f(x), f(x) = f′(x) 34 f then the value of is____ 1/2a)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous such that R→ R If d2(f−1(x)) is an increasing function from exists then fx> 0andf−1 'f' 35 is b. c. d. cannot be determined < 0 > 0 = 0 dx2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous 3 1 + t 3 2 dy dy 36 if satisfies then is: f(x) ⋅ { } x = , y = + = 1 + f(x) t3 2t2 t dx dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let

  11. 2 f(x) = { ,ξsrationalb, 37 1 + x2 ξsrational has exactly two points of continuity then the value of are b. c. d. (0, 3] [0, 1] (0,2] b φ  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous [cos( )] πx a − x 2a πx If for and for then f(x) = {sin( )tan[ ] x > a x < a, 38 2 2a a − x b. has a removable discontinuity at f f(a+) < 0 c. has an irremovable x = a f f(a−) < 0 discontinuity at d. x = a  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let αcotx β f(x) = { + ,0 < |x| x2 x 39 1 ≤ 1 , x = 0 3 If is continuous at f(x) then the value of is b. c. d. 9 1 2 α2+ β2 x = 0 5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Statement 1: Minimum number of points of discontinuity of the function f(x) = (g(x)[2x − 1]∀x ∈ ( − 3, − 1) 40 , where [.] denotes the greatest integer function and Statement 2: can be continuous at a point of discontinuity, say if [2x − 1] g(c1) = 0. is zero. g(x) = ax3+ x2+ 1 of f(x) x = c1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous π π Let , (where [.] denotes the greatest integer less f(x) = [tanx[cotx]],x[ ] , 12 12 41 than or equal to ). Then the number of points, where b. zero c. three d. infinite is discontinuous is a. one f(x) x  Watch Free Video Solution on Doubtnut

  12. CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let and that be any function which is such that is iirrational for rational x, then in [a,b] f(x) is rational for irrational x f:[a,b] → R f(x) 42  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous 100 ∏ n=1 f(101) (x − n)n(101−n) 43 If ; then f(x) = = f'(101)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous A x = t + t3andy = t2, wheret ∈ R 1 1 2 curve in the xy-plane is parametrically given by is the parameter. For what value(s) of is t 44 dy b. c. d. = ? 3 1 2 3 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous

  13. Number of points of discontinuity of (where [.] denotes the greatest integer function) a. 0 b. 1 c. 2 d. 3 in its domain is equal to f(x) = [sin−1x] − [x] 45  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If g(x) xmf(x) + h(x) + 3 46 = ( lim )m− → x ≠ 1andg(1) = e3 ∞ 2xm+ 4x + 1 when at x = 1 such that are continuous functions f(x), g(x)andh(x) then the value of is b. c. d. 7 6 5f(1) − 2h(1) 9 8  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous ∣∣∣ is continuously differentiable function with ∣∣∣= 0 f'(x) f(x) Suppose f''(x) f'(x) 47 f(x) − 1 and satisfies and then is f′(x) ≠ 0 f(0) = 1 f'(0) = 2 ( lim )x→0 x b. c. d. 1 1/2 2 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous (x − a)4 (x − a)3 (x − a)4 (x − a)3 ∣ ∣∣∣ ∣ ∣ ∣∣∣ ∣ ∣ ∣∣∣ ∣ ∣ ∣∣∣ ∣ 1 1 If then . (x − b)4 (x − b)3 (x − b)4 (x − b)3 f(x) = f'(x) = λ 1 1 48 (x − c)4 (x − c)3 (x − c)4 (x − c)3 1 1 Find the value of λ

  14.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let f(x + y) − f(x) f(y) − a = 2 2 + xy for all real If is differentiable and √5a − 1 − a2 exists for all real permissible f′(0) 49 f(x) xandy. value of and is equal to negative for all real Then is positive for all real f(x) is x f(x) a . has real roots Nothing can be said about the sign of x f(x) = 0 f(x)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous satisfies the equation f:R− − − → 1,∞ A function f(xy) = f(x)f(y) − f(x) − f(y) If differentiable on + 2. R − {0}andf(2) = 5, f′(x) 50 . f(x) − 1 = λthenλ = x 1 b. c. d. 2′f(1) 3f′(1) f′(1) f′(1) 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous be a continuous function and f:R→ R Let is true If f(x) = f(2x) ∀x ∈ R. 51 1 then the value of is equal to 6 (b) 0 (c) (d) ∫ f(1) = 3, f(f(x))dx 3f(3) 2f(0) − 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous g(x) Let and If g(0) = g′(0) = 0andg0= 17 f(x) = whenx ≠ 0 f(0) = 0. 52 x then b. c. d. f(0) = 3/4 −1/2 17/3 17/2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND

  15. DIFFERENTIABILITY_Miscellaneous Let f(x + y) = f(x) + f(y) + 2xy 53 − 1 for all real that f(x) > 0∀x ∈ R. and be a differentiable function. If the prove f′(0) = cosα, f(x) xandy  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous f:( − ∞,∞)− − − → Let be a continuous function such that 0,∞ f(x + y) = f(x) + f(y) 54 + f(x)f(y), ∀x ∈ R. f'(0) = 1. 8 Also c. d. 6 7 Then equal represents the greatest integer function b. [f(2)] ([.] ) 5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous −1k,x 1 f(x) = {(x2+ e ) 2−x 55 = 2,x ≠ 2 is continuous from right at the point of these then equals b. k c. d. none x = 2, 0 1/4 −1/4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous The x = 2t = |t|,y = t3+ t2|t|a = 0 derivative of the function is a. -1 b. 1 c. 0 d. does not exist represented parametrically as 56

  16.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If and then show that y = secnθ − cosnθ x = secθ − cosθ 2 57 dy (x2+ 4)( = n2(y2+ 4) ) dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If cos x 1 − cos(1 − ) f(x) = { 2 1x 58 2mxn = 0,x ≠ 0 f(0) = 1 and is continuous at then the value of x = 0 is a. 2 b. 3 c. -3 d. 7 m + n  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous 1 Which of the following functions is/are discontinuous at x = 1? f(x) = 1 + 2tanx 1 g(x) = ( lim )x− → ∞ 1 + n ∈ s2(πx) 1 h(x) = 2−2^ ((( ))), x 59 1 − x ≠ 1andh(1) = 1 x − 1 φ(x) = ,x |x − 1| + 2(x − 1)2 ≠ 1andφ(1) = 1  Watch Free Video Solution on Doubtnut

  17. CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If t(1 + x2) = xandx2+ t2 60 = ythenatx = 2, 24 101 488 358 dy the value of is b. c. d. 5 125 125 125 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous f:R→ A function f(x) is defined as R 61 ax2+ bx + c + enx = ( lim )n− → ∞ 1 + c.enx is continuous on then find the values of a,b,c. R  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If d x + y = 3e2the (xy) 62 dx = 0f or x = e2 ee b. c. d. e 2e2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous

  18. 63 dy y and then n= 1 b. 2 c. 3 d. 4 xy = (x + y)n = dx x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If π f(x) = {sin( )(x − [x]), x 2 < 55(b − 1), x ab2∣∣x2− 11x + 24∣∣ = 5 ,x > 5 x − 3 64 is continuous at 25 then ([.] denotes the greatest integer function) 17 a = , b = 1 2 36 x = 5, a,b ∈ R 6 a = 6 25 23 6 b. c. d. a = ,b = , b = a = ,b = 108 5 13 29 100 5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If 2cosx − sin2x f(x) = { ,x (π − 2x)2 e−cotx− 1 65 π π ≤ , x > 8x − 4π 2 2 , then which of the following holds? (a) is continuous at irremovable discontinuity at (d)None of these x = π/2 (b) has an f x = π/2 f (c) has a removable discontinuity at x = π/2 f  Watch Free Video Solution on Doubtnut

  19. CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let 1 f(x) = {8 ,x < 0a[x], a ∈ R x 66 − {0}, x ≥ 0, (where [.] denotes the greatest integer function). Then (a) a finite number of points (b)Discontinuous at a finite number of points. (c)Discontinuous at an infinite number of points. (d)Discontinuous at is Continuous only at f(x) x = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous let f(x) = x3− x2− 3x − 1,g(x) = (x + 1)a f(x) and where is a rational function such that it is continuous 67 h(x) = (1) h g(x) everywhere except when and x = − 1,(2) lim x→ ∞h(x) = ∞ x→ −1h(x) =1 then the value of (3) lim h(1) 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous For which of the following functions exists such that f(0) sin|x| is continuous at x=0 f(x) 1 π b. c. d. f(x) = (cos( )) f(x) = xsin( ) f(x) = 68 (log)e|x| 1 x x f(x) = 1 + 2cotx  Watch Free Video Solution on Doubtnut

  20. CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let 1 − sinπx f(x) = [ , x 1 + cos2πx 1 69 < and p, x 2 √2x − 1 1 = and 2 √4 + √2x − 1 − 2 x =1 .Determine the value of p, if possible, so that the function is continuous at . 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let be any function. Also is defined by for all f:R → R g:R → R g(x) = |f(x)| Then is a. Onto if is onto b. One-one if is one-one c. Continuous if is continuous d. None of these 70 x. f f f  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If f(x) a sin(2x2) 3x = {( )) + cos( b 71 a b /x2,x ≠ 0e3, x = 0 is continuous at d. 0 −1/2 then minimum value of is b. c. x = 0∀b ∈ R −1/8 −1/4 a  Watch Free Video Solution on Doubtnut

  21. CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous f:R→ Let be continuous functions satisfying 72 f(x) R Then the value of is b. c. d. 5 2 3 f(0) = 1andf(2x) − f(x) = x. f(3) 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous f:R→ . . ′(x2) x The function satisfies for all real Given that x. f(x2) = f′(x) 73 R f f and , then the value of is b. c. d. 8 2 4 f1= 8 f′(1) + f1 f(1) = 1 6  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous xasinbx (lim)x→ , wherea,b, c sin(xc) 0 74 ∈ R~{0},eξstsandhas non-zero value. Then, (b) (d) none of these −1 0 a + c = b  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous −1 −1 −3 −2 d2y d2y d2y d2x dy dy equals: (b) (d) ( ) −( ) ( ) ( )( ) dy2 dx2 −3 dx2 dx2 dx dx 75 d2y dy −( )( ) dx2 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If f(1) = 3, f′(1) = 2,f1= 4, 76 ' '(3) = then(f−1) −1 a. 1 b. c. -2 d. none of these 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND

  22. DIFFERENTIABILITY_Miscellaneous 2 d2y dy 77 y , Prove that = x x3 = (x − y) (a + bx)e x dx2 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous 3/2 2] dy [1 + ( ) dx If , then can be put in the form of R2/3 R = d2y dx2 1 1 1 1 2 2 b. c. 78 + − + 2/3 2/3 2/3 2/3 2/3 2/3 d2y d2y d2y d2x d2x d2x ( ) ( ) ( ) ( ) ( ) ( ) dy2 dy2 dy2 dx2 dx2 dx2 . 1 1 d. 2/3 2/3 d2y d2x ( ) ( ) dy2 dx2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous d2y 3 3 3 3t 79 If (b) (c) (d) x = t2y = t3,then = dx2 2 2 (4t) 2(t)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND

  23. DIFFERENTIABILITY_Miscellaneous x3 80 The function in does not take the value [ − 4,4] b. f(x) = − sinπx + 4 −4 8 c. d. 10 18 12  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous The number of points of discontinuity of `f(x)={1+x ,0lt=xlt=2 3-x ,22` where is defined as, f(x) g(x) = f(f(x)) 81  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous (128a + ax)1/8− 2 If the function is continuous at , then the value x = 0 f(x) = (32 + bx)1/5− 2 64 82 3 of is b. c. d. none of these f(0) 28/5f(0) a/b f(0) 5 5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let f(x) = ( lim )x− → ∞ n−1 ∑ x 83 (rx + 1){(r + 1)x + 1} r=0 . Then not differentiable at is a periodic function. f(x) is continuous but not differentiable at x = 0 f(x) is both continuous but f(x) x = 0 f(x) is neither continuous not differentiable at x = 0  Watch Free Video Solution on Doubtnut

  24. CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let f(x) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ = n cot(πx) + (px2+ 2) ax(x − 1)( ) 4 lim n→ ∞ , n cot (πx) 84 ( ) + 1 4 x ∈ (0,1) ∪ (1,2) and 0,x = 1 If f(x) is differentiable for all then equals (a2+ p2) x ∈ (0, 2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous If f(x) = {sinx, x ≠ nπ,n ∈ I2, otherwise g(x) = {x2+ 1,x ≠ 0,4,x 85 = 05, x = 2 (lim)x→ then 0g{f(x)}is =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous

  25. 1 The number of points at which is not differentiable, where g(x) = 2 1 + 86 f (x) 1 , is b. c. d. 1 2 f(x) = 3 4 1 1 + x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous f(x) = x1/3(x − 2)2/3 If for all d. then the domain of is b. x, x ∈ R − {0} f' 87 c. {x ∣ x⟩0} x ∈ R − {0,2} x ∈ R  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let be a function with continuous second derivative and f(0) = f′(0) = 0. f f(x) Determine a function by g Then which of the g(x) = { ,x ≠ 00, x = 0 x 88 following statements is correct? has a continuous first derivative has a first derivative is continuous but fails to have a derivative has a first derivative but the first derivative is not continuous g g g g g  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND DIFFERENTIABILITY_Miscellaneous Let continuous at be a function defined on with Assume that is f(x) ( − a, a) a > 0. f(x) f(x) − f(kx) x = 0and( lim )x→ 89 0 x = α, wherek ∈ (0, 1) α then b. c. is differentiable at f(x) d. is f′(0+) = 0 f′(0−) = x = 0 f(x) 1 − k non-differentiable at x = 0  Watch Free Video Solution on Doubtnut  Download Doubtnut to Ask Any Math Question By just a click  Get A Video Solution For Free in Seconds  Doubtnut Has More Than 1 Lakh Video Solutions  Free Video Solutions of NCERT, RD Sharma, RS Aggarwal, Cengage (G.Tewani), Resonance DPP, Allen, Bansal, FIITJEE, Akash, Narayana, VidyaMandir

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