Question

Let $f\left(x\right)$ be a quadratic expression which is positive $\forall x\in R$ . Then, the value of $10\left[f\right(x)+f(-x\left)\right]$ is

• A. $>0$
• B. $\ge 0$
• C. $<0$
• D. $\le 0$

Answer 1

The correct option is A $>0$

$f\left(x\right)=a{x}^2+bx+c$ is positive for all real $x$, then $a>0$ and $D<0$
Then, $f\left(0\right)>0$ so $c>0$
So, we have $g\left(x\right)=10\left(f\right(x)+f(-x\left)\right)$
$g\left(x\right)=10(a{x}^2+bx+c+(a{x}^2-bx+c\left)\right)$
$g\left(x\right)=10(2a{x}^2+2c)$
$g\left(x\right)=20(a{x}^2+c)$ as $a>0$ and its $D=-80ac$ as $c>0$
So, we get $D<0$
So, $g\left(x\right)$ is always positive.