Q109 / 225 · JEE Main 2004 · Mathematics

If $2 a+3 b+6 c=0$, then at least one root of the equation $a x^2+b x+c=0$ lies in the interval

  • Correct answer
    A
    $(0,1)$
  • B
    $(1,2)$
  • C
    $(2,3)$
  • D
    $(1,3)$

Solution

Let $f^{\prime}(x)=a x^2+b x+c \Rightarrow f(x)=\frac{a x^3}{3}+\frac{b x^2}{2}+c x+d$ $\Rightarrow f(x)=\frac{1}{6}\left(2 a x^3+3 b x^2+6 c x+6 d\right)$, Now $f(1)=f(0)=d$, then according to Rolle's theorem $\Rightarrow f^{\prime}(x)=a x^2+b x+c=0$ has at least one root in $(0,1)$

Hence, the correct answer is option: 'A'.

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