Rolle’s Theorem MCQ

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• Find the value of ‘a’ & ‘b’ if f(x) = ax2 + bx + sin(x) is continuous over and differentiable over (0, π) and satisfy the Rolle’s theorem at point c = π⁄4.
  ক

  and differentiable over (0, π) and satisfy the Rolle’s theorem at point c = π⁄4. ] 0.45,1.414

  খ

  0, π

  গ

  0, π

  ঘ

  0, π
• Find the value of ‘a’ if f(x) = ax2+32x+4 is continuous over and differentiable over (-4, 0) and satisfy the Rolle’s theorem. Hence find the point in interval (-2,0) at which its slope of a tangent is zero
  ক

  and differentiable over (-4, 0) and satisfy the Rolle’s theorem. Hence find the point in interval (-2,0) at which its slope of a tangent is zero ] 2, -2

  খ

  -4, 0

  গ

  -4, 0

  ঘ

  -4, 0
• f(x) = 3Sin(2x), is continuous over interval and differentiable over interval (0,π) and c ∈(0,π)
  ক

  and differentiable over interval (0,π) and c ∈(0,π) ] π

  খ

  0,π

  গ

  0,π

  ঘ

  0,π
• Find value of c where f(x) = sin(x) ex tan(x), c ∈ (0,∞) c) Tan-1 d) Rolle’s Theorem is not applied, Cannot find the value of c
  ক

  Tan-1 c) Tan-1 d) Rolle’s Theorem is not applied, Cannot find the value of c View Answer]

  খ

  -(2+c2)/(1+c2) [ Tan-1[-(2-c2)/(1+c2)

  গ

  -(2+c2)/(1+c2) b) Tan-1[-(2-c2)/(1+c2)

  ঘ

  -(2+c2)/(1+c2) b) Tan-1[-(2-c2)/(1+c2)
• Find the value of c if f(x) = sin3(x)cos(x), is continuous over interval and differentiable over interval (0, π⁄2) and c ∈(0, π⁄2)
  ক

  and differentiable over interval (0, π⁄2) and c ∈(0, π⁄2) ] 0

  খ

  0, π⁄2

  গ

  0, π⁄2

  ঘ

  0, π⁄2
• Find the value of c if f(x) = x(x-3)e3x, is continuous over interval and differentiable over interval (0, 3) and c ∈(0,3)
  ক

  and differentiable over interval (0, 3) and c ∈(0,3) ] 0.369

  খ

  0,3

  গ

  0,3

  ঘ

  0,3
• Find the value of c(a point where slope of a atangent to curve is zero) if f(x) = Sin(x) is continuous over interval and differentiable over interval (0, π) and c ∈(0,π)
  ক

  and differentiable over interval (0, π) and c ∈(0,π) ] π

  খ

  0,π

  গ

  0,π

  ঘ

  0,π
• Rolle’s theorem is applicable to the and continuous in open interval (a, b) only and having same value at point ‘a’ and ‘b’ b) Functions continuous in closed interval only and having same value at point ‘a’ and ‘b’ c) Functions continuous in closed interval and differentiable in open interval (a, b) only and having same value at point ‘a’ and ‘b’ d) Monotonically Increasing funtions
  ক

  Functions differentiable in closed interval and continuous in open interval (a,

  খ

  a, b

  গ

  a, b

  ঘ

  a, b
• Rolle’s Theorem is a special case of
  ক

  Lebniz Theorem

  খ

  Mean Value Theorem

  গ

  Taylor Series of a function

  ঘ

  Leibnit’x Theorem
• Rolle’s Theorem tells about the
  ক

  Existence of point c where derivative of a function becomes zero

  খ

  Existence of point c where derivative of a function is positive

  গ

  Existence of point c where derivative of a function is negative

  ঘ

  Existence of point c where derivative of a function is either positive or negative
• For second degree polynomial it is seen that the roots are equal. Then what is the relation between the Rolles point c and the root x?
  ক

  c = x

  খ

  c = x2

  গ

  They are independent
• For all second degree polynomials with y = ax2 + bx + k, it is seen that the Rolles’ point is at c = 0. Also the value of k is zero. Then what is the value of b?
  ক

  0

  খ

  1

  গ

  -1
• Let f(x) = x + sin(x) Every point on the graph is rotated by 45 degree with respect to the origin along the radius equal to the radius vector at that point. How many c that belong to exist Such that f'(c) = 0.
  ক

  exist Such that f'(c) = 0. ] 10

  খ

  0, 11π

  গ

  0, 11π

  ঘ

  0, 11π
• For y = -x2 + 2x there exist a c in the interval Such that f'(c) = 0.
  ক

  Such that f'(c) = 0. ] True

  খ

  - 19765, 19767

  গ

  - 19765, 19767

  ঘ

  - 19765, 19767