Question

'$af\left(k\right)<0$' is the necessary and sufficient condition for a particular real number $k$ to lie between the roots of a quadratic equation $f\left(x\right)=0$, where $f\left(x\right)=a{x}^2+bx+c$. If $f\left(k_1\right)f\left(k_2\right)<0$, then exactly one of the roots will lie between $k_1$ and $k_2$.

If $c(a+b+c)<0<a(a+b+c)$, then

• A. both the roots lie in $(0,1)$
• B. one root is less than $0$, the other is greater than $1$
• C. one root lies in $(0,1)$ and the other in $(1,\infty )$
• D. one root lies in $(-\infty ,0)$ and the other in $(0,1)$

Answer 1

The correct option is D one root lies in $(-\infty ,0)$ and the other in $(0,1)$

$c(a+b+c)<0<a(a+b+c)$
$f\left(0\right)f\left(1\right)<0$
So, one root lie between $0$ and $1$.
$af\left(1\right)>0$
So, $1$ doesn't lie between the roots.

Hence, one root lies in $(-\infty ,0)$ and the other in $(0,1)$.