The minimum integral value of $\alpha$ for which the quadratic equation $\left({\cot }^{-1}\alpha \right)x^2-{\left({\tan }^{-1}\alpha \right)}^{3/2}x+2{\left({\cot }^{-1}\alpha \right)}^2=0$ has both positive roots

$\left({\cot }^{-1}\alpha \right)x^2-{\left({\tan }^{-1}\alpha \right)}^{3/2}x+2{\left({\cot }^{-1}\alpha \right)}^2=0$

Equation has real roots

$\therefore D\ge 0\Rightarrow {\left({\tan }^{-1}\alpha \right)}^3-8{\left({\cot }^{-1}\alpha \right)}^3\ge 0$

$\therefore {\tan }^{-1}\alpha \ge 2{\cot }^{-1}\alpha =\pi -2{\tan }^{-1}\alpha$

$\Rightarrow {\tan }^{-1}\alpha \ge \frac{\pi }{3}\Rightarrow \alpha \ge \sqrt{3}$

Sum of roots $>0$

$\Rightarrow \frac{{\left({\tan }^{-1}\alpha \right)}^{3/2}}{2{\cot }^{-1}\alpha }>0$

${\tan }^{-1}\alpha >0\Rightarrow \alpha >0$

Product of roots $>0$

$\Rightarrow 2{\cot }^{-1}\alpha >0\Rightarrow \alpha \in R$

$\Rightarrow \alpha \ge \sqrt{3}$.