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Manipal MET2020MathematicsApplication of Derivatives

If 2 a+3 b+6 c=0 , then the equation a x^2+b x+c=0 has atleast one real root in

Options

  1. A(0,1)
  2. B(0, 1 2 )
  3. C( 1 4 , 1 2 )
  4. DNone of these

Correct answer

A. (0,1)

Step-by-step solution

Let f(x)= 1 3 a x^3+ 1 2 b x^2+c x Then, f(x) is a polynomial. So, it is continuous in R . Now, f(0)=0 and f(1)= a 3 + b 2 +c= 2 a+3 b+6 c 6 f(1)=0 [ 2 a+3 b+6 c=0 , given ] f(x) is a polynomial, so it is differentiable in R ; so in (0,1) . Hence, by Rolle's theorem, there exists atleast one point x (0,1) , there exists such that f^ (x)=0 a x^2+b x+c=0 Hence, required interval is (0,1) .

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