Question

Let $f\left(x\right)$ be a quadratic expression which is positive for all real values of x. If $g\left(x\right)$ = $f\left(x\right)$ + $f‘\left(x\right)$ + $f^{\prime \prime }\left(x\right)$, then for any real x

• A. $g\left(x\right)<0$
• B. $g\left(x\right)>0$
• C. $g\left(x\right)=0$
• D. $g\left(x\right)\ge 0$

Answer 1

Answer: (B)

$g\left(x\right)>0$

Consider $f\left(x\right)=x^2+x+1$ which is positive for all x and does not cut the x axis at any real point.
Hence
$f\left(x\right)=x^2+x+1$ ...(i)
$f^{\prime }\left(x\right)=2x+1$ ...(ii)
$f^{\prime \prime }\left(x\right)=2$ ...(iii)

Hence Summing i ii and iii we get
$g\left(x\right)=x^2+3x+3$.
Hence
$g\left(x\right)$ is of the form $a{x}^2+bx+c$ where $a,b,c>0$ and
$b^2-4ac=D<0$
Hence
$g\left(x\right)>0$ for $xϵR$.