If $f(x) = \begin{cases} e^x + ax, & x < 0 \\ b(x - 1)^2, & x \ge 0 \end{cases}$ is differentiable at $x = 0$,then $(a, b)$ is

If $f(x) = \begin{cases} x[x], & 0 \le x < 2 \\ (x-1)[x], & 2 \le x \le 4 \end{cases}$,where $[.]$ denotes the greatest integer function,then:

The function $f(x) = |\cos x|$ is

Let $K$ be the set of all real values of $x$,where the function $f(x) = \sin |x| - |x| + 2(x - \pi) \cos |x|$ is not differentiable. Then the set $K$ is

Consider the following statements.
$(a)$ If a function is differentiable at a point $p$ then it is not continuous at $p$.
$(b)$ If a function is not continuous at $x = a$, then it is not differentiable at $x = a$.
$(c)$ If $f(x) = |x|$ then $f(x)$ is not differentiable but continuous on $R$.
$(d)$ If $f(x) = x - [x]$, then $f'(1) = 1$.
Which of the above statements are (is) correct?

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 $(b^2 - 2a + c^6)$.