Question

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

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

Answer 1

The correct option is B $g\left(x\right)\ge 0$

Let $f\left(x\right)=a{x}^2+bx+c>0\forall x\in R$
$\Rightarrow b^2-4ac<0$ and $a>0$ ...(1)
Now $g\left(x\right)=f\left(x\right)-f^{\prime }\left(x\right)+f^{\prime \prime }\left(x\right)=(a{x}^2+bx+c)-(2ax+b)+2a$
$=a{x}^2+(b-2a)x-(2a-b+c)$
$\Rightarrow \Delta ={(b-2a)}^2-4a(2a-b+c)=(b^2-4ac)-4a^2<0$ (Using (1))
$\Rightarrow g\left(x\right)>0\forall x\in R\Rightarrow g\left(x\right)\ge 0\forall x\in R$