Question

If m and $M$ are the least and the greatest values of ${\left(co{s}^{-1}x\right)}^2+{\left(si{n}^{-1}+x\right)}^2$, then $\frac{M}{m}=$

• A. $10$
• B. $5$
• C. $4$
• D. $2$

Answer 1

Answer: (A)

${\left(\frac{\pi }{2}-si{n}^{-1}x\right)}^2+{\left(si{n}^{-1}x\right)}^2$
$=\frac{{\pi }^2}{8}+2{\left[si{n}^{-1}x-\frac{\pi }{4}\right]}^2$
$m=\frac{{\pi }^2}{8}$,
$M=\frac{5{\pi }^2}{4}$,
$\frac{M}{m}=10$