Question

Let $f :\left[\frac{1}{2}, \infty\right) \rightarrow\left[\frac{3}{4}, \infty\right)$ defined by $f(x)=x^2-x+1$ and $x_0$ be the value of $x$ satisfying $f(x)=f^{-1}(x)$. Then the value of $\sin ^{-1}\left(x_0\right)+\cot ^{-1}\left(-x_0\right)$ equals

• A. $ \frac{5 \pi}{4}$
• B. $\frac{\pi}{4}$
• C. $\frac{3 \pi}{4}$
• D. none

Answer 1

Answer: (A)

$f ( x )= f ^{-1}( x )$
$\Rightarrow x _0=1 . \text { Now } \sin ^{-1}(1)+\cot ^{-1}(-1)=\frac{\pi}{2}+\frac{3 \pi}{4}=\frac{5 \pi}{4}$