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Application of Derivatives

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  1. 1.Let f:R R be a differentiable function such that f((x+y)/(3))=(f(x)+f(y))/(3) for all x,y …
  2. 2.Let f(x) be a polynomial of degree 5, and have extrema at x=1 and x=-1.
  3. 3.The number of critical points of the function f(x)= array|c|c| |( x)/(x)|, & x 0 1, & x=0 …
  4. 4.The least value of ( ^2 -6 +3 ^2 +2) is
  5. 5.Let and respectively be the maximum and the minimum values of the function f( )=4( ^4((7 )…
  6. 6.Let f(x)=x^2025-x^2000, x [0,1] and the minimum value of the function f(x) in the interval…
  7. 7.Let f:R R be a twice differentiable function such that f^''(x)>0 for all x R and f^'(a-1)=…
  8. 8.Let f : R [ ] R be a polynomial function of degree four having extreme values at x = 4 and…
  9. 9.Let x=-1 and x=2 be the critical points of the function f(x)=x^3+ax^2+b _e|x|+1,x 0.
  10. 10.Let a>0.
  11. 11.The shortest distance between the curves y^2=8 x and x^2+y^2+12 y+35=0 is:
  12. 12.Let f:R R be a function defined by f(x)=||x+2|-2|x .
  13. 13.If the function f(x)=2x^3-9ax^2+12a^2x+1, where a>0, attains its local maximum and local m…
  14. 14.The sum of all local minimum values of the function f(x)=array|c|c| 1-2x, & x2 array is
  15. 15.Let (2,3) be the largest open interval in which the function f(x)=2 _e(x-2)-x^2+ax+1 is st…
  16. 16.Consider the region R=(x,y):x y 9-(11)/(3)x^2,x 0.
  17. 17.A spherical chocolate ball has a layer of ice-cream of uniform thickness around it.
  18. 18.Let f(x)= _x^2^0t^2-8t+15e^tdt,x R.
  19. 19.If the function f(x)=2x^3-9ax^2+12a^2x+1,a>0 has a local maximum at x= and a local minimum…
  20. 20.Let f(x)=4 ^3x+3√3 ^2x-10.
  21. 21.The number of critical points of the function f(x)=(x-2)^2 / 3(2 x+1) is
  22. 22.For the function f(x)=( x)-x+1,x R, between the following two statements (S1) f(x)=0 for o…
  23. 23.The interval in which the function f(x)=x^x, x>0, is strictly increasing is
  24. 24.Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the…
  25. 25.Let f(x)=x^5+2x^3+3x+1,x R, and g(x) be a function such that g(f(x))=x for all x R.
  26. 26.For the function f(x)= x+3x-(2)/( )(x^2+x),x [0,( )/(2)], consider the following two state…
  27. 27.Let f(x)=3x-2+√4-x be a real valued function.
  28. 28.Let the sum of the maximum and the minimum values of the function [f(x)=(2 x^2-3 x+8)/(2 x…
  29. 29.If 5f(x)+4f((1)/(x))=x^2-2, 0 and y=9x^2f(x), then y is strictly increasing in :
  30. 30.Let f: R (0,) be strictly increasing function such that x lim(f(7x))/(f(x))=1.
  31. 31.If the function f:(-,-1] (a,b] defined by f(x)=e^x^3-3x+1 is one - one and onto, then the …
  32. 32.f(x)=vmatrix x^3 2x^2+1 1+3x 3x^2+2 2x x^3+6 x^3-x 4 x^2-2 vmatrixx R2f(0)+f^'(0)
  33. 33.Let [f(x)=(x+3)^2(x-2)^3, x [-4,4]].
  34. 34.The maximum area of a triangle whose one vertex is at (0,0) and the other two vertices lie…
  35. 35.The function f(x)=xx^2-6x-16,x R--2,8
  36. 36.The function f(x)=2x+3(x)^(2)/(3),x R, has
  37. 37.Consider the function f:[(1)/(2),1] R defined by f(x)=4√2x^3-3√2x-1.
  38. 38.Let g(x)=3f((x)/(3))+f(3-x) and f^''(x)>0 for all x (0,3).
  39. 39.0 x maxx-2 x x+(1)/(3) 3x=
  40. 40.If the local maximum value of the function f(x)=(√3e2 x)^ ^2x,x (0,( )/(2)) , is (k)/(e), …
  41. 41.Let f:[2,4] R be a differentiable function such that (x _ex)f^'(x)+( _ex)f(x)+f(x) 1,x [2,…
  42. 42.Let g(x)=f(x)+f(1-x) and f^''(x)>0,x (0,1).
  43. 43.The slope of tangent at any point (x, y) on a curve y=y(x) is x^2+y^22xy,x>0.
  44. 44.A square piece of tin of side 30 cm is to be made into a box without top by cutting a squa…
  45. 45.If the functions f(x)=x^33+2bx+ax^22 and g(x)=x^33+ax+bx^2,a 2b have a common extreme poin…
  46. 46.The number of real solutions of x^7 + 5x^3 + 3x + 1 = 0 is equal to ____________.
  47. 47.For the function [f(x) = 4 _e(x - 1) - 2x^2 + 4x + 5,,x > 1], which one of the following i…
  48. 48.The number of real roots of the equation e^4x + 2e^3x - e^x - 6 = 0 is :
  49. 49.A wire of length 20 m is to be cut into two pieces.
  50. 50.The height of a right circular cylinder of maximum volume inscribed in a sphere of radius …