Question

If $M$ denotes the maximum value of $(1+{\sec }^{-1}x)(1+{\cos }^{-1}x)$ & $m$ denotes the maximum value of $(1+cose{c}^{-1}x)(1+{\sin }^{-1}x)$, then $\left[\frac{M}{m}\right]$ is (where [.] denotes greatest integer function)

Answer 1

$(1+{\sec }^{-1}x)(1+{\cos }^{-1}x)$ its
Maximum value occur at $x=-1$
$\Rightarrow M=(1+\pi {)}^2$
$(1+cose{c}^{-1}x)(1+{\sin }^{-1}x)$
its maximum value occur at $x=1$
$\Rightarrow m=(1+\frac{\pi }{2}{)}^2$
$\Rightarrow \left(\frac{M}{m}\right)=4.{\left(\frac{1+\pi }{2+\pi }\right)}^2=2.6$

$\therefore \left[\frac{M}{m}\right]=2$