Question

If the function $x^2+ax+1$ is increasing in $(1,2)$, then smallest possible value of a is ________.

• A. $1$
• B. $-2$
• C. $2$
• D. $-1$

Answer 1

Answer: (B)

$-2$

For an increasing function $f^{\prime }\left(x\right)>0$
$\therefore f^{\prime }\left(x\right)=2x+a>0$
$\therefore \forall x\in (1,2)$ we have $f^{\prime }\left(x\right)>0$
Now $1<x<2$ (definition of interval)
$\therefore 2<2x<4$
$\therefore 2+a<2x+a<4+a$
$\therefore 2+a\ge 0\Rightarrow a\ge -2$ then $f^{\prime }\left(x\right)\ge 0$
$\therefore$ Minimum value of a is $-2$.