Let $y=\left({\cot }^{-1}x\right)\left({\cot }^{-1}\right(-x\left)\right)$ and range of $y$ is $(0,\frac{a{\pi }^2}{b}]$, where $a,b$ are coprime numbers, then the value of $a+b$ is

The correct option is D $5$
Given : $y=\left({\cot }^{-1}x\right)\left({\cot }^{-1}\right(-x\left)\right)$
$\Rightarrow y={\cot }^{-1}x[\pi -{\cot }^{-1}(x\left)\right]$
As ${\cot }^{-1}\left(x\right)>0,(\pi -{\cot }^{-1}(x\left)\right)>0$

Using A.M. $\ge$ G.M., we get
$\frac{{\cot }^{-1}x+(\pi -{\cot }^{-1}(x\left)\right)}{2}\ge \sqrt{\left({\cot }^{-1}x\right)[\pi -{\cot }^{-1}(x\left)\right]}\Rightarrow \frac{\pi }{2}\ge \sqrt{{\cot }^{-1}\left(x\right)(\pi -{\cot }^{-1}(x\left)\right)}>0\Rightarrow \frac{{\pi }^2}{4}\ge y>0\Rightarrow a=1,b=4\therefore a+b=5$