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Limits, Continuity and Differentiability

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  1. 251.If f(x)=( x) and f''(x)+ x,f'(x)+g(x)=0, then g(x) is:
  2. 252.If f:RR is a function defined by f(x)=[x](2x-12), where [x] denotes the greatest integer f…
  3. 253.Let f:[1,3]R be a function satisfying x[x] f(x)√6-x for all x2, and f(2)=1, where [x] is t…
  4. 254.Statement 1: A function f:RR is continuous at x_0 if and only if _x x_0f(x) exists and _x …
  5. 255.Consider the function f(x)=|x-2|+|x-5|, xR.
  6. 256.If f(x)=a| x|+be^|x|+c|x|^3, where a,b,cR, is differentiable at x=0, then
  7. 257.If x+|y|=2y, then y as a function of x, at x=0 is
  8. 258.If f'(x)=( x) and y=f(2x+33-2x), then dydx equals
  9. 259.Consider a quadratic equation ax^2+bx+c=0, where 2a+3b+6c=0, and let g(x)=ax^33+bx^22+cx.
  10. 260.Define f(x) as the product of two real functions f_1(x)=x, xR, and f_2(x)=cases1x, & x00, …
  11. 261.The values of p and q for which the function f(x)=cases(p+1)x+ xx, & x<0, & x=0x+x^2- xx^3…
  12. 262.If function f(x) is differentiable at x=a, then _x ax^2f(a)-a^2f(x)x-a is:
  13. 263.d^2xdy^2 equals:
  14. 264.Let f:(-1,1)R be a differentiable function with f(0)=-1 and f'(0)=1.
  15. 265.Let y be an implicit function of x defined by x^2x-2x^x y-1=0.
  16. 266.Let f(x)=cases(x-1)1x-1, & x10, & x=1cases.
  17. 267.The function f:R0R given by f(x)=1x-2e^2x-1 can be made continuous at x=0 by defining f(0)…
  18. 268.Let f:RR be a function defined by f(x)=x+1,|x|+1.
  19. 269.A value of c for which the conclusion of the Mean Value Theorem holds for the function f(x…
  20. 270.If x^m y^n=(x+y)^m+n, then dydx is:
  21. 271.If f is a real valued differentiable function satisfying |f(x)-f(y)|(x-y)^2, x,yR, and f(0…
  22. 272.Suppose f(x) is differentiable at x=1 and _h01hf(1+h)=5, then f'(1) equals
  23. 273.Let f be differentiable for all x.
  24. 274.If the equation a_nx^n+a_n-1x^n-1++a_1x=0 (a_10, n2) has a positive root x=, then the equa…
  25. 275.Let f(x)=1- x4x-, x4, x(0,2).
  26. 276.If x=e^y+e^y+ to , x>0, then dydx is
  27. 277.If 2a+3b+6c=0, then at least one root of the equation ax^2+bx+c=0 lies in the interval
  28. 278.If f(x)=casesxe^-((1)/(|x|)+1x), & x00, & x=0cases, then f(x) is
  29. 279.Let f(x) be a polynomial function of second degree.
  30. 280.If f(x)=x^n, then the value of f(1)-f'(1)1!+f''(1)2!-f'''(1)3!++(-1)^nf^(n)(1)n! is
  31. 281.Let f(a)=g(a)=k and their nth derivatives f^(n)(a),g^(n)(a) exist and are not equal for so…
  32. 282.f is defined on [-5,5] as f(x)=x if x is rational, =-x if x is irrational.
  33. 283.If f(x+y)=f(x) f(y) x,y and f(5)=2, f'(0)=3, then f'(5) is
  34. 284.If y=(x+√1+x^2)^n, then (1+x^2)d^2ydx^2+xdydx is
  35. 285.If 2a+3b+6c=0, (a,b,cR) then the quadratic equation ax^2+bx+c=0 has
  36. 286.[ _x ( x^2 + 5x + 3 x^2 + x + 2 )^x]
  37. 287.Let f(2) = 4 and f'(x) = 4. Then [ _x 2 xf( 2 ) - 2f( x ) x - 2] is given by
  38. 288.[ _x 0 x^n - [ x ] [ x ]], [n N], ( [x] denotes the greatest integer less than or equal to…
  39. 289.If [f( 1 ) = 1,f'( 1 ) = 2,] then [ _x 1 f( x ) - 1 x - 1] is
  40. 290.f is defined in [[ - 5,5 ]] as [f( x ) = x] if x is rational [,,,,,,,,,,,,,] = - x if x is…
  41. 291.f(x) and g(x) are two differentiable functions on [0, 2] such that f''(x) - g''(x) = 0, f'…
  42. 292.If [ _x 0 ( 3 + x ) - ( 3 - x ) x] = k, the value of k is
  43. 293.[ _x 2 [ 1 - ( x 2 ) ][ 1 - x ] [ 1 + ( x 2 ) ][ - 2x ]^3] is
  44. 294.If [ _x ( 1 + a x + b x^2 )^2x = e^2], then the value of a and b, are
  45. 295.Let [] and [] be the distinct roots of ax^2 + bx + c = 0, then [ _x 1 - ( ax^2 + bx + c ) …
  46. 296.Suppose f(x) is differentiable at x = 1 and [ _h 0 1 hf( 1 + h ) = 5], then [f'( 1 )] equa…
  47. 297.The function [f:R/ 0 R] given by [f( x ) = 1 x - 2 e^2x - 1] can be made continuous at x =…
  48. 298.Let [f:R R] be a positive increasing function with [ _x f(3x) f(x) = 1].
  49. 299.[ _x 2 ( 1 - 2(x - 2) x - 2 )]
  50. 300.[ _x 0 ( 1 - 2x )( 3 + x ) x 4x] is equal to