Question

Let $f:R\to R$ be defined as $f\left(x\right)=max\{\left|{\tan }^{-1}x\right|,{\cot }^{-1}x\}.$ Consider the following statements :
$\text{I.}$ $f\left(x\right)$ is continuous and differentiable for all $x\in R$.
$\text{II.}$ Range of $f$ is $[\frac{\pi }{4},\pi ]$.
$\text{III.}$ $f\left(x\right)$ is many-one and into.

Which of the following is TRUE ?

• A. All three statements are wrong
• B. Exactly one of above statements is correct
• C. Exactly two of above statements are correct
• D. All three statements are correct

Answer 1

The correct option is B Exactly one of above statements is correct




From the above graphs,
Range $:[\frac{\pi }{4},\pi )$
$f\left(x\right)$ is many-one and into
and $f\left(x\right)$ is not differentiable at $x=1$

Statements $\text{I}$ and $\text{II}$ are wrong.