Question

If $R$ is the least value of $a$ such that the function $f\left(x\right)=x^2+ax+1$ is increasing on $[1,2]$ and $S$ is the greatest value of $a$ such that the function $f\left(x\right)=x^2+ax+1$ is decreasing on $[1,2]$ then the value of $|R-S|$ is

Answer 1

Given: $f\left(x\right)=x^2+ax+1$
$f^{\prime }\left(x\right)=2x+a$
For increasing function $2x+a{|}_{x=1}\ge 0$
$a\ge -2$
For decreasing function $2x+a{|}_{x=2}\le 0$
$a\le -4$
$R=-2,S=-4$
$|R-S|=2$