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MHT CET202619 April 2026Evening ShiftMathematicsApplication of DerivativesActual

Rolle's theorem holds for monic quadratic polynomial f(x) on the interval [ , + 3] where f( ) = 0 . Similarly, g(x) = f(x) + 2 also follows Rolle's theorem in the interval [ , 3] where g(3) = 0 , such that the value of c is the same for both f(x) and g(x) . Then the value of (f g)( ) is...

Options

  1. A-4
  2. B4
  3. C-2
  4. D2

Correct answer

C. -2

Step-by-step solution

Since f(x) is a monic quadratic polynomial and Rolle's theorem holds on [ , + 3] with f( ) = 0 , we have f( + 3) = f( ) = 0 . Thus, f(x) = (x - )(x - - 3) . The value of c for f(x) is the midpoint of the interval, c = + 3 2 . For g(x) = f(x) + 2 , Rolle's theorem holds on [ , 3] . The value of c is the same, so the midpoint of [ , 3] is also + 3 2 . + 3 2 = + 3 2 = 2 . We are given g(3) = 0 , which means f(3) + 2 = 0 f(3) = -2 . Substituting x = 3 in f(x) , we get: (3 - )(3 - - 3) = -2 - (3 - ) = -2 ^2 - 3 + 2 = 0

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