Question

Consider $f\left(x\right)=a{x}^2+bx+c$ with $a>0,$
If both roots of the quadratic equation are smaller than any constant $k,$ then

• A. $f\left(k\right)>0$
• B. None of these
• C. $-\frac{b}{2a}>k$
• D. $f\left(k\right)<0$

Answer 1

The correct option is A $f\left(k\right)>0$

Let, $\alpha ,\beta$ be the roots of $f\left(x\right)=a{x}^2+bx+c=0$

When both roots of $f\left(x\right)=0$ are smaller than $k,$


From the graph,

$D>0$ when roots are real and distinct, and $D=0$ when roots are real and equal.

So, we can express it as $D\ge 0$

$x-$ coordinate of vertex $\Rightarrow -\frac{b}{2a}<k$

And $f\left(k\right)>0$

Thus, the required conditions are

$\left(i\right)D\ge 0$
$\left(ii\right)-\frac{b}{2a}<k$
$\left(iii\right)f\left(k\right)>0$