Question

Let $f (x)$ be a quadratic expression which is positive for all real values of $x$. If $g(x) = f (x) + f ' (x) + f " (x), $ then for any real $x$

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

Answer 1

Answer: (B)

Let $ f \, (x) = ax^2 + bx + c > 0, \, \forall x \, \in R $
$\Rightarrow a > 0 $
and $ b^2 - 4 ac < 0 $ ...(i)
$\therefore g(x) = f (x) + f'(x ) + f '' (x) $
$\Rightarrow g (x) = ax^2 + bx + c + 2ax + b + 2a $
$\Rightarrow g (x) = ax^2 + x \, (b + 2a) + (c + b + 2a)$
whose discriminant
= $ (b + 2a)^2 - 4a \, (c + b + 2a)$
= $ b^2 + 4a^2 + 4ab - 4ac - 4ab - 8a^2 $
= $ b^2 - 4a^2 - 4ac = ( b^2 - 4ac) < 0 $ [ from Eq. (i) ]
$\therefore \, g \, (x) > 0, \, \forall \, x, $ as $a > 0$ and discriminant $< 0 $.
Thus, $g (x) > 0, \forall \, x \, \in R$.