Question

For the expression $f\left(x\right)=a{x}^2+bx+c,(a>0),$ having both real roots, the condition for both real roots to be greater than a real value $x_0$ is

• A. $f\left(x_0\right)>0,x_0=\frac{-b}{2a}$
• B. $f\left(x_0\right)\ge 0,x_0<\frac{-b}{2a}$
• C. $f\left(x_0\right)>0,x_0<\frac{-b}{2a}$
• D. $f\left(x_0\right)<0,x_0\ge \frac{-b}{2a}$

Answer 1

The correct option is C

$f\left(x_0\right)>0,x_0<\frac{-b}{2a}$

As $a>0$, graph is cup shaped parabola.

Consider the graph of $f\left(x\right)$ in which roots of $f\left(x\right)$ is shown to be greater than $x_0$.

We know $f\left(x_0\right)>0$.

Now, for $x_0$ to be on left of $\alpha$ & $\beta$, and also $f\left(x_0\right)>0$, we can comfortably take $x_0<\frac{-b}{2a}$.

So, $A$ is right option.