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MHT CET202525 Apr 2025Morning ShiftMathematicsApplication of DerivativesActual

If Rolle's theorem holds for the function x^3+a x^2+b x, 1 x 2 at the point 4 3 , then the values of a and b are respectively.

Options

  1. A5,8
  2. B-8,5
  3. C8,-5
  4. D-5,8

Correct answer

D. -5,8

Step-by-step solution

Rolle's Theorem requires that f(x) is continuous on [1,2] , differentiable on (1,2) , and satisfies f(1) = f(2) . Since f(x) = x^3 + a x^2 + b x is a polynomial, the continuity and differentiability conditions are automatically satisfied. Equating f(1) = 1 + a + b and f(2) = 8 + 4a + 2b gives 1 + a + b = 8 + 4a + 2b , which simplifies to 3a + b = -7 . The derivative is f'(x) = 3x^2 + 2a x + b . Given that f' ( 4 3 ) = 0 , we have 3 ( 4 3 )^2 + 2a ( 4 3 ) + b = 0 , which becomes 16 3 + 8a 3 + b = 0 . Multiplying thr

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