Thank you for reporting, we will resolve it shortly
Q.
If $2 a+3 b+6 c=0$, then atleast one root of the equation $a x^{2}+b x+c=0$ lies in the interval
ManipalManipal 2016
Solution:
Consider the function $f(x)=\frac{a x^{3}}{3}+\frac{b x^{2}}{2}+c x+d$
We have, $f(0)=d$
and $f(1)=\frac{a}{3}+\frac{b}{2}+c+d=\frac{2 a+3 b+6 c}{6}+d$
$=\frac{0}{6}+d[\because 2 a+3 b+6 c=0]$
Therefore, $0$ and $1 $ are roots of the polynomial $f(x)$.
Hence, according to Rolle's theorem, there exists atleast one root of the polynomial $f'(x)=a x^{2}+b x+c$ lying between $0$ and $1 .$