If $f(x)=\sqrt{x+\sin x}$,then all the points of the set $\{(x, f(x)) \mid f^{\prime}(x)=0\}$ lie on

The function $f(x) = x^{3} - 6x^{2} + ax + b$ is such that $f(2) = f(4) = 0$. Consider two statements.
$(S_1)$ There exists $x_{1}, x_{2} \in (2, 4)$,$x_{1} < x_{2}$,such that $f^{\prime}(x_{1}) = -1$ and $f^{\prime}(x_{2}) = 0$.
$(S_2)$ There exists $x_{3}, x_{4} \in (2, 4)$,$x_{3} < x_{4}$,such that $f$ is decreasing in $(2, x_{4})$,increasing in $(x_{4}, 4)$ and $2f^{\prime}(x_{3}) = \sqrt{3}f(x_{4})$.
Then

Let $f: R \rightarrow R$ be defined by $f(x) = \frac{x^2-3x-6}{x^2+2x+4}$. Then which of the following statements is (are) $TRUE$?
$(A)$ $f$ is decreasing in the interval $(-2, -1)$
$(B)$ $f$ is increasing in the interval $(1, 2)$
$(C)$ $f$ is onto
$(D)$ Range of $f$ is $[-\frac{3}{2}, 2]$

Let $f(x) = \min (\{x\}, \{e^{-x}\})$ for $x \in [0, 10]$. If $C$ and $D$ are the number of points where $f(x)$ is discontinuous and non-differentiable respectively,then $(C + D)$ is equal to (where $\{.\}$ denotes the fractional part function).

$f(x)=4 \log _{e}(x-1)-2 x^{2}+4 x+5, x>1$,which one of the following is $NOT$ correct?

Let $f(x) = x \cos^{-1}(-\sin |x|)$,$x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. Then which of the following is true?