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JEE Advanced2010MathematicsApplication of DerivativesActual

Paragraph: Consider the polynomial f(x)=1+2 x+3 x^2+4 x^3 . Let s be the sum of all distinct real roots of f(x) and let t=|s| . Question: The real number s lies in the interval

Options

  1. A(- 1 4 , 0 )
  2. B(-11,- 3 4 )
  3. C(- 3 4 ,- 1 2 )
  4. D(0, 1 4 )

Correct answer

C. (- 3 4 ,- 1 2 )

Step-by-step solution

Given, f(x)=4 x^3+3 x^2+2 x+1 aligned & f^ (x)=2 (6 x^2+3 x+1 ) & D=9-24 0 f (- 3 4 ) & =1- 6 4 + 27 16 - 108 64 & = 64-96+108-108 64 < 0 aligned f(x) changes its sign in (- 3 4 , -1 2 ) , hence f(x)=0 has a root in ( -3 4 , -1 2 ) .

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