Question

Consider $f\left(x\right)=a{x}^2+bx+c$ with $a>0$,
If both roots of the quadratic equation are greater than any constant $k$. The necessary and sufficient condition for this are :

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

Answer 1

Answer: (A), (C), (D)

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


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

When both roots of $f\left(x\right)$ are greater than $k.$


From graph,

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

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

$x$-coordinates of vertex, i.e, $-\frac{b}{2a}>k$

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

So, required conditions are

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