Question

where $\left[\right.\cdot \left]\right.$ is the greatest integer function. If $f$ is a continuous function, then $ab=\frac{\sqrt{\lambda }}{\mu }$ . Find value of $\lambda \mu $ (where $\lambda ,\mu $ are coprime to each other)

Answer 1

$f(x)=\left\{\begin{array}{ll}\frac{a+3 \cos x}{x^{2}}, & x<0 \\ b \tan \left(\frac{\pi}{[x+3]}\right), \quad x \geq 0\end{array}\right.$
$f\left(0^{-}\right)=\frac{-3}{2}, \quad a=-3$
$f\left(0^{+}\right)=\sqrt{3} b, \quad b=\frac{-\sqrt{3}}{2}$