If $f(x) = \begin{cases} x^{3}-3x+2, & x < 2 \\ x^{3}-6x^{2}+9x+2, & x \geq 2 \end{cases}$,then:

Let $f(x) = (\sin(\tan^{-1} x) + \sin(\cot^{-1} x))^2 - 1$ for $|x| > 1$. If $\frac{dy}{dx} = \frac{1}{2} \frac{d}{dx}(\sin^{-1}(f(x)))$ and $y(\sqrt{3}) = \frac{\pi}{6}$,then $y(-\sqrt{3})$ is equal to

Let $f:[0, \infty) \rightarrow [0, 3]$ be a function defined by
$f(x) = \begin{cases} \max \{\sin t : 0 \leq t \leq x\}, & 0 \leq x \leq \pi \\ 2 + \cos x, & x > \pi \end{cases}$
Then which of the following is true?

Let $f(x) = \lim_{n}$ ${\rightarrow \infty} \left( \frac{n^n(x+n)(x+\frac{n}{2}) \cdots (x+\frac{n}{n})}{n!(x^2+n^2)(x^2+\frac{n^2}{4}) \cdots (x^2+\frac{n^2}{n^2})} \right)^{\frac{x}{n}}$,for all $x > 0$. Then
$(A)$ $f(\frac{1}{2}) \geq f(1)$
$(B)$ $f(\frac{1}{3}) \leq f(\frac{2}{3})$
$(C)$ $f^{\prime}(2) \leq 0$
$(D)$ $\frac{f^{\prime}(3)}{f(3)} \geq \frac{f^{\prime}(2)}{f(2)}$

Let $f(x) = \begin{cases} 3x, & x < 0 \\ \min \{1+x+[x], x+2[x]\}, & 0 \leq x \leq 2 \\ 5, & x > 2 \end{cases}$ where $[.]$ denotes the greatest integer function. If $\alpha$ and $\beta$ are the number of points where $f$ is not continuous and is not differentiable,respectively,then $\alpha + \beta$ equals.......

Consider the following three statements for the function $f : (0, \infty) \rightarrow \mathbb{R}$ defined by $f(x) = |\log_{e} x| - |x - 1|$:
$(I)$ $f$ is differentiable at all $x > 0$.
$(II)$ $f$ is increasing in $(0, 1)$.
$(III)$ $f$ is decreasing in $(1, \infty)$.
Then: