Question

If the expression $f\left(x\right)=x^2+6x+k$ is non negative $\forall x\in R$, then the interval in which $k$ lies is

• A. $(-\infty ,\infty )$
• B. $(9,\infty )$
• C. $[9,\infty )$
• D. $(-\infty ,9)$

Answer 1

The correct option is C $[9,\infty )$

$f\left(x\right)=x^2+6x+k$,
$a=1$, $b=6$, $c=k$
As $f\left(x\right)$ is always $\ge 0$, so graph touches or lies above the $x-$axis.
Hence, $D\le 0$
$\Rightarrow b^2-4ac\le 0$
$\Rightarrow 36-4k\le 0$
$\Rightarrow 4k-36\ge 0$
$\Rightarrow k\ge 9\Rightarrow k\in [9,\infty )$