Question

Let $f\left(x\right)$ be a polynomial of degree two which is positive for all $x\in R$. lf $g\left(x\right)=f\left(x\right)+f^{\prime }\left(x\right)+f^{\prime \prime }\left(x\right)+x{f}^{\prime \prime \prime }\left(x\right)+x^2{f}^{iv}\left(x\right)$, 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)

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

$f\left(x\right)$ is a polynomial of degree $=2$
$f\left(x\right)>0\forall x\in R$
$f^{{}^{\prime \prime \prime }}\left(x\right)=0$ and $f^{iv}\left(x\right)=0$
Let $f\left(x\right)=a{x}^2+bx+c$
$f^{{}^{\prime }}\left(x\right)=2ax+b$
$f^{{}^{\prime \prime }}\left(x\right)=2a$
$g\left(x\right)=a{x}^2+(2a+b)x+(c+b+2a)$
Now, discriminant of $g\left(x\right)=(b+2a{)}^2-4a(b+c+2a)$
$=-4a^2+(b^2-4ac)<0$ ........ $\left[f\right(x)$ is positive for all real $x]$
Therefore, $g\left(x\right)>0$ for all real $x$.