If R is the least value of a such that the function f x   =   x 2   +   a x   +…

If $f^{\text{'}}\left(x\right)=2x+a\ge 0\forall x\in [1,2] \Rightarrow a\ge -2x\forall x\in [1,2]$

$a_{\min }=R=-2$ If

$a_{max}=S=-4$

$|R-S|=2$

Alternate :

If $f\left(x\right)$ is increasing for all $x\in [1,2]$ then

$-\frac{a}{2}\le 1\Rightarrow a\ge -2\Rightarrow R=-2$

If $f\left(x\right)$ is decreasing for $x\in [1,2],$ then

$-\frac{a}{2}\ge 2\Rightarrow a\le -4\Rightarrow S=-4$

$|R-S|=2$