The domain of the derivative of the functions $f(x)\left\{\begin{array}{cc}\tan ^{-1} x, & \text { if }|x|…
The domain of the derivative of the functions $f(x)\left\{\begin{array}{cc}\tan ^{-1} x, & \text { if }|x| \leq 1 \\ \frac{1}{2}(|x|-1), & \text { if }|x|>1\end{array}\right.$ is given by
$R-\{1\}$
$R-\{0\}$
$R-\{-1,1\}$
$R-\{-1\}$
Solution
$\begin{aligned} & f(x) \begin{cases}\frac{1}{2}(-x-1) & x < -1 \\ \tan ^{-1} x & -1 \leq x \leq 1 \\ \frac{1}{2}(x-1) & x > 1\end{cases} \\ & \Rightarrow f^{\prime}(x) \begin{cases}-\frac{1}{2} & x < -1 \\ \frac{1}{1+x^2} & -1 < x < 1 \\ \frac{1}{2} & x > 1\end{cases} \end{aligned}$
$\because f(x)$ discontinuous at $x=-1$ and $x=1$
Hence $f(x)$ is not differentiable at $x=-1$ and $x=1$