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

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

Let $f(x)=2^x-x^2, x \in R$. If $m$ and $n$ are respectively the number of points at which the curves $y=f(x)$ and $y=f^{\prime}(x)$ intersect the $x$-axis,then the value of $m+n$ is

If $f(x)=\int_0^x e^{t^2}(t-2)(t-3) dt$ for all $x \in(0, \infty)$,then
$(A)$ $f$ has a local maximum at $x=2$
$(B)$ $f$ is decreasing on $(2,3)$
$(C)$ there exists some $c \in(0, \infty)$ such that $f^{\prime \prime}(c)=0$
$(D)$ $f$ has a local minimum at $x=3$

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: