Question

Let $f(x) = \begin{cases} \cot^{-1}x, |x| \ge 1 \\ \frac{1}{2}|x| + \frac{\pi}{4} - \frac{1}{2}, |x| < 1 \end{cases}$ then number of values of x, ($x \in R$) which do not belong to the domain of $f'(x)$ is

Answer 1

3

Answer 2

The function $f(x)$ is analyzed for differentiability at points where its definition changes ($x=-1, x=1$) and where the absolute value function changes its behavior ($x=0$).

• At $x=-1$, the function is discontinuous, hence not differentiable.
• At $x=0$, the left and right derivatives of the piecewise linear part are different ($-\frac{1}{2}$ and $\frac{1}{2}$), hence not differentiable.
• At $x=1$, the function is continuous, but the left and right derivatives are different ($\frac{1}{2}$ and $-\frac{1}{2}$), hence not differentiable. Thus, $f'(x)$ is undefined at $x=-1, 0, 1$.