Question

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

Answer 1

Answer: 2

$f(x)=x^2+ax+1$
$f^{\prime }(x)=2x+a$
when $f(x)$ is increasing on $[1,2]$
$2x+a\ge 0\forall x\in [1,2]$
$a\ge -2x\forall x\in [1,2]$
$R=-4$
when $f(x)$ is decreasing on $[1,2]$
$2x+a\le 0\forall x\in [1,2]$
$a\le -2\forall x\in [1,2]$
$S=-2$
$|R-S|=|-4+2|=2$