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

Two differentiable functions $f(x)$ and $g(x)$ are such that $f''(x) > 0$ and $g''(x) < 0$ for all $x \in (a,b)$ and $\int_{a}^{b} f(x) dx = \int_{a}^{b} g(x) dx$. If $f(x) = g(x)$ for $x = \alpha, \beta \in (a,b)$ $(\alpha < \beta)$,then:

Let $f(x) = \sin x + (x^3 - 3x^2 + 4x - 2) \cos x$ for $x \in (0, 1)$. Consider the following statements:
$I.$ $f$ has a zero in $(0, 1)$.
$II.$ $f$ is monotone in $(0, 1)$.
Then,

For the function $f(x) = x^4 (12 \ln x - 7)$,which of the following statements is true?

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 $[-\frac{3}{2}, 2]$

If $(x-a)^{2}+(y-b)^{2}=c^{2},$ for some $c > 0,$ prove that $\frac{\left[1+\left(\frac{d y}{d x}\right)^{2}\right]^{\frac{3}{2}}}{\frac{d^{2} y}{d x^{2}}}$ is a constant independent of $a$ and $b.$