The number of points at which the function,$f(x) = |x - 0.5| + |x - 1| + \tan x$ does not have a derivative in the interval $(0, 2)$ is :

If $f(x)=|x|+|\sin x|$ for $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$,then its left hand derivative at $x=0$ is

Let $f(x) = \begin{cases} a \cot^{-1} \left( \frac{b+x}{4} \right), & \frac{-2}{3} < x < 0 \\ 2, & x = 0 \\ \frac{\ln(1-cx)}{x}, & 0 < x < \frac{2}{3} \end{cases}$. If the function $f(x)$ is differentiable at $x = 0$,then find the value of $(b^2 - 2a + c^6)$.

If $f(x) = \begin{cases} x^2 \left| \cos \frac{\pi}{x} \right|, & x \neq 0 \\ 0, & x = 0 \end{cases}$, then at $x = 2$, $f(x)$ is

Let $g(x) = x \cdot f(x)$,where $f(x) = \begin{cases} x \sin \frac{1}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases}$. Discuss the differentiability of $g$ at $x = 0$.

The function $y = |\sin x|$ is continuous for any $x$,but it is not differentiable at: