Question

Consider $f\left(x\right)=a{x}^2+bx+c$ with $a>0$,
If exactly one root of the quadratic equation $f\left(x\right)=0$ lies between $k_1$ and $k_2$ where $k_1<k_2$. The necessary condition for this is:

• A. $a.f\left(k_2\right)<0$
• B. $-\frac{b}{2a}>k_1$
• C. $f\left(k_1\right).f\left(k_2\right)>0$
• D. $D<0$

Answer 1

The correct option is B $-\frac{b}{2a}>k_1$

Let, $\alpha ,\beta$ be the roots of $f\left(x\right)=a{x}^2+bx+c$

Graph for condition, exactly one root of the quadratic equation $f\left(x\right)$ lies between $k_1$ and $k_2$.


From graph, required conditions,
$\left(i\right)-\frac{b}{2a}>k_1$
$\left(ii\right)D>0$
$\left(iii\right)f\left(k_1\right)\cdot f\left(k_2\right)\le 0$