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Application of Derivatives

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  1. 151.The value of c in the Lagrange's mean value theorem for the function f(x) = x3 - 4x2 + 8x …
  2. 152.The length of the perpendicular from the origin, on the normal to the curve, x2 + 2xy -- 3…
  3. 153.A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness t…
  4. 154.Let f be a real valued function, defined on R - -1, 1 and given by f(x) = 3 loge [| x - 1 …
  5. 155.The function f(x) = x^3 - 6x^2 + ax + b is such that f(2) = f(4) = 0.
  6. 156.The sum of all the local minimum values of the twice differentiable function f : R [] R de…
  7. 157.Let 'a' be a real number such that the function f(x) = ax2 + 6x - 15, x [] R is increasing…
  8. 158.For which of the following curves, the line [x + 3 y = 2 3 ] is the tangent at the point […
  9. 159.If the curve y = ax2 + bx + c, x[ ]R, passes through the point (1, 2) and the tangent line…
  10. 160.The function f(x) = [4x^3 - 3x^2 6 - 2 x + ( 2x - 1 ) x] :
  11. 161.If the tangent to the curve y = x3 at the point P(t, t3) meets the curve again at Q, then …
  12. 162.If Rolle's theorem holds for the function f(x) = x^3 - ax^2 + bx - 4, [x [1,2]] with [f'( …
  13. 163.Let [f(x) = 3 ^4x + 10 ^3x + 6 ^2x - 3], [x [ - 6, 2 ]].
  14. 164.The local maximum value of the function [f(x) = ( 2 x )^x^2], x > 0, is
  15. 165.Let slope of the tangent line to a curve at any point P(x, y) be given by [xy^2 + y x].
  16. 166.The maximum slope of the curve [y = 1 2x^4 - 5x^3 + 18x^2 - 19x] occurs at the point :
  17. 167.Let f be any function defined on R and let it satisfy the condition : [|f(x) - f(y)|, ,|(x…
  18. 168.A box open from top is made from a rectangular sheet of dimension a [] b by cutting square…
  19. 169.The slope of normal at any point (x, y), x > 0, y > 0 on the curve y = y(x) is given by [x…
  20. 170.Let [] ^ * be the largest value offor which the functionf_ (x) = 4 x^3 - 36 x^2 + 36x + 48…
  21. 171.The surface area of a balloon of spherical shape being inflated, increases at a constant r…
  22. 172.If the tangent at the point (x1, y1) on the curve y = x^3 + 3x^2 + 5 passes through the or…
  23. 173.The sum of absolute maximum and absolute minimum values of the function [f(x) = |2x^2 + 3x…
  24. 174.Let [ x - 2y = ] be a tangent to the hyperbola a^2x^2 - y^2 = b^2.
  25. 175.If the absolute maximum value of the function [f(x)=(x^2-2 x+7) e^(4 x^3-12 x^2-180 x+31)]…
  26. 176.The curve y(x)=a x^3+b x^2+c x+5 touches the x-axis at the point [P(-2,0)] and cuts the y-…
  27. 177.Water is being filled at the rate of 1 cm3 / sec in a right circular conical vessel (verte…
  28. 178.If the angle made by the tangent at the point (x0, y0) on the curve [x = 12(t + t t)], [y …
  29. 179.If the maximum value of a, for which the function [f_a(x)= ^-1 2 x-3 a x+7] is non-decreas…
  30. 180.Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r.
  31. 181.Let S be the set of all the natural numbers, for which the line [x a + y b = 2] is a tange…
  32. 182.Let [f(x) = 2 ^ - 1x + 4 ^ - 1x - 3x^2 - 2x + 10], [x [ - 1,1]].
  33. 183.The function [f(x)=x e^x(1-x), x R], is :
  34. 184.If the minimum value of [f(x)=5 x^22+x^5, x>0], is 14 , then the value of [] is equal to :
  35. 185.Let [f(x)=3^(x^2-2)^3+4, x R].
  36. 186.Let f : R [] R be a function defined by f(x) = (x - 3)n1 (x - 5)n2, n1, n2 [] N.
  37. 187.A wire of length 22 m is to be cut into two pieces.
  38. 188.If xy4 attains maximum value at the point (x, y) on the line passing through the points (5…
  39. 189.Let [f(x) = 4x^3 - 11x^2 + 8x - 5,,x R].
  40. 190.If the local maximum value of the function [f(x)=(√3 e2 x)^ ^2 x, x (0, ()/(2))] , is [(k)…
  41. 191.[ _0 x x-2 x x+(1)/(3) 3 x=]
  42. 192.The sum of the absolute maximum and minimum values of the function [f(x)=|x^2-5 x+6|-3 x+2…
  43. 193.Let the function [f(x) = 2x^3 + (2p - 7)x^2 + 3(2p - 9)x - 6] have a maxima for some value…
  44. 194.Let x=2 be a local minima of the function [f(x)=2x^4-18x^2+8x+12,x(-4,4)].
  45. 195.Let [f:(0,1)R] be a function defined [f(x) = 1 1 - e^ - x], and [g(x) = ( f( - x) - f(x) )…
  46. 196.If the functions f(x)=(x^3)/(3)+2 b x+(a x^2)/(2) and g(x)=(x^3)/(3)+a x+b x^2, a 2 b have…
  47. 197.The number of points on the curve [y=54 x^5-135 x^4-70 x^3+180 x^2+210 x] at which the nor…
  48. 198.A wire of length [20 ~m] is to be cut into two pieces.
  49. 199.If 5 f(x)+4 f((1)/(x))=x^2-2, x 0 and y=9 x^2 f(x), then y is strictly increasing in :
  50. 200.Let [g(x)=3 f((x)/(3))+f(3-x)] and [f^ (x)>0] for all [x (0,3)].