Question

If $f\left(x\right)={\left({\sin }^{-1}x\right)}^2+{\left({\cos }^{-1}x\right)}^2$, then

• A. $f\left(x\right)$ has the least value of $\frac{{\pi }^2}{8}$
• B. $f\left(x\right)$ has the greatest value of $\frac{5{\pi }^2}{8}$
• C. $f\left(x\right)$ has the least value of $\frac{{\pi }^2}{16}$
• D. $f\left(x\right)$ has the greatest value of $\frac{5{\pi }^2}{4}$

Answer 1

Answer: (A), (D)

The correct options are
A $f\left(x\right)$ has the least value of $\frac{{\pi }^2}{8}$
D $f\left(x\right)$ has the greatest value of $\frac{5{\pi }^2}{4}$

The least value for the above function occurs at $x=\frac{1}{\sqrt{2}}$
$\Rightarrow$
$=\frac{{\pi }^2}{16}+\frac{{\pi }^2}{16}$
$=\frac{{\pi }^2}{8}$ ...(i)
The greatest value for the above function occurs at $x=-1$
$\Rightarrow$
$=\frac{{\pi }^2}{4}+{\pi }^2$
$=\frac{5{\pi }^2}{4}$ ...(ii)