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MHT CET202525 Apr 2025Evening ShiftMathematicsApplication of DerivativesActual

The function f (x)=x^3-6 x^2+a x+b satisfyies the conditions of Rolle's theorem in [1,3] . Then the values of a and b are respectively

Options

  1. A11,-6
  2. B-6,11
  3. C-11,6
  4. D6, -11

Correct answer

A. 11,-6

Step-by-step solution

Rolle's Theorem requires f(x) to be continuous on [1,3] , differentiable on (1,3) , and satisfy f(1) = f(3) . Since f(x) = x^3 - 6x^2 + ax + b is a polynomial, continuity and differentiability are automatic. Equating f(1) = f(3) : 1 - 6 + a + b = 27 - 54 + 3a + b a - 5 = 3a - 27 2a = 22 a = 11 The only option with a=11 is A, implying b=-6 . With a=11 , the derivative becomes f'(x) = 3x^2 - 12x + 11 . Solving f'(x)=0 yields x = 2 3 3 1.423, 2.577 , both in (1,3) . The values satisfying all conditions are A .

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