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MHT CET202615 April 2026Morning ShiftMathematicsApplication of DerivativesActual

If the function f(x) = ax^3 - bx^2 - 8x - 4 satisfies Roll's theorem in [1,3] , if f'(2) = 0 then a - b is equal to...

Options

  1. A2
  2. B-2
  3. C0
  4. D1

Correct answer

A. 2

Step-by-step solution

Since f(x) satisfies Rolle's theorem in [1, 3] , we have f(1) = f(3) . a(1)^3 - b(1)^2 - 8(1) - 4 = a(3)^3 - b(3)^2 - 8(3) - 4 a - b - 12 = 27a - 9b - 28 26a - 8b = 16 13a - 4b = 8 Also, f'(x) = 3ax^2 - 2bx - 8 . Given f'(2) = 0 , we have: 3a(2)^2 - 2b(2) - 8 = 0 12a - 4b = 8 Subtracting the second equation from the first: (13a - 4b) - (12a - 4b) = 8 - 8 a = 0 Substituting a = 0 into 12a - 4b = 8 : -4b = 8 b = -2 Therefore, a - b = 0 - (-2) = 2 .

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