Question

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

Answer 1

$f(x)=x^{2}+a x+1 $
$f'(x)=2 x+a$
when $f(x)$ is increasing on $[1,2]$
$2 x + a \geq 0 \quad \forall x \in[1,2] $
$a \geq-2 x \,\,\,\forall x \in[1,2]$
$R =-4$
when $f(x)$ is decreasing on $[1,2]$
$2 x + a \leq 0 \,\,\,\forall x \in[1,2]$
$a \leq-2 \,\,\,\forall x \in[1,2] $
$S =-2 $
$| R - S |=|-4+2|=2$