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 and sufficient condition(s) for this are :

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

Answer 1

Answer: (B), (C), (D)

$-\frac{b}{2a}>k_1$

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,
(i) $D>0$ and,
(ii) $f\left(k_1\right).f\left(k_2\right)<0$

and check the solution in case if any of $k_1,k_2$ is one of the root or not.