Question

Let $f:R\to R,f\left(x\right)=max.\{|{\tan }^{-1}x|,{\cot }^{-1}x\}$. Consider the following statements:
I. Function is continuous and derivable $\forall x\in R$
II. Range of function $\left[\frac{\pi }{4},\pi \right]$
III. $f\left(x\right)$ is many one-into
Identify the correct option:

• A. All 3 statements are wrong.
• B. Exactly one of the above statement is correct.
• C. Exactly two of the above statement is correct.
• D. All 3 statement are correct.

Answer 1

The correct option is A All 3 statements are wrong.

The above Figure is the graph of Function $f\left(x\right)$

Here we can see that at $x=\frac{\pi }{4}$ $f\left(x\right)$ is Continuous but not Differentiable,so statement (I) is Wrong

The range starts from $\frac{\pi }{4}$ to $\pi$ but not touches $\pi$,so statement (II) is Wrong

$f\left(x\right)$ is one-one function so, statement (III) is wrong

$\therefore$ Ans is $Opt:\left[A\right]$