Question

Find the sign of the quadratic polynomial.
$f\left(x\right)=x^2+5|x|+6$

• A. $y>0\forall xϵ(-\infty ,2)\cup (3,\infty )y<0\forall xϵ(2,3)y=0\forall xϵ2,3$
• B. $y>0\forall xϵ(-\infty ,-3)\cup (-2,2)\cup (3,\infty )y<0\forall xϵ(-3,-2)\cup (2,3)y=0\forall xϵ\{-3,-2,2,3\}$
• C. $y>0\forall xϵR$
• D. $y>0\forall xϵ(-\infty ,-3)\cup (-2,\infty )y<0\forall xϵ(-3,-2)y=0\forall xϵ\{-3,-2\}$

Answer 1

The correct option is C $y>0\forall xϵR$

Here, equation is $x^2+5|x|+6.$
First term is $x^2$, which is a perfect square and hence is always non negative. Second term is $5|x|$, where $|x|$ will always be positive. And third term is 6, which is positive.
$\Rightarrow x^2+5|x|+6$ will always be positive $\Rightarrow y>0\forall xϵR$