Question

Suppose that $f\left(x\right)$ is a quadratic expression 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. $g\left(x\right)>0$
• C. $g\left(x\right)=0$
• D. $g\left(x\right)\ge 0$

Answer 1

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

Let $f\left(x\right)=a{x}^2-bx+cf\left(x\right)>0$

for every $x\in R$ then,$a>0$ and$b^2-4ac<0g\left(x\right)=f\left(x\right)+f^{{}^{\prime }}\left(x\right)+f^{{}^{\prime \prime }}\left(x\right)g\left(x\right)=a{x}^2+x(b+2a)+(b+2a+c)$

Discriminant of $g\left(x\right)$ is $D={(b+2a)}^2-4a(b+2a+c)=b^2-4ac-4a^2$

As$b^2-4ac<0\therefore b^2-4ac-4a^2<0\therefore g\left(x\right)>0$

for all $x\in R$