Let $g:R\to R$ be a differentiable function with $g\left(0\right)=0, g^{\text{'}}\left(0\right)=0$ and $g^{\text{'}}\left(1\right)\ne 0$. Let $f\left(x\right)=\left\{\begin{array}{cc}\frac{x}{\left|x\right|}g\left(x\right),& x\ne 0\\ 0,& x=0\end{array}\right.$ and $h\left(x\right)=e^{\left|x\right|}$ for all $x\in R.$ Let $\left(foh\right)\left(x\right)$ denotes $f\left\{h\left(x\right)\right\}$ and $\left(hof\right)\left(x\right)$ denotes $h\left\{f\left(x\right)\right\}$. Then, which of the following is/are true?

Let $f_1:R\to R,f_2:,\left(-\frac{\pi }{2},\frac{\pi }{2}\right)\to R,f_3:\left(-1,e^{\frac{\pi }{2}}-2\right)\to R$ and $f_4:R\to R$ be functions defined by

(i) $f_1\left(x\right)=\sin \left(\sqrt{1-e^{-x^2}}\right)$

(ii) $f_2\left(x\right)=\left\{\begin{array}{ccc}\frac{\left|\sin x\right|}{{\tan }^{-1}x}& \text{ if }& x\ne 0\\ 1& \text{ if }& x=0\end{array}\right.$ where the inverse trigonometric function ${\tan }^{-1}x$ assumes values in $\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$

(iii) $f_3\left(x\right)=\left[\sin \left({\log }_e(x+2)\right)\right],$ where for $t\in R,\left[t\right]$ denotes the greatest integer less than or equal to$t$

(iv) $f_4\left(x\right)=\left\{\begin{array}{cc}{x}^2\sin \left(\frac{1}{x}\right)& \text{ if }x\ne 0\\ 0& \text{ if }x=0\end{array}\right.$

The value of $p$ and $q$ for which the function $$\begin{gathered} f\left(x\right)=\left\{\begin{array}{l}\begin{array}{cc}\frac{\sin (p+1)x+\sin x}{x},& x<0\end{array} \\ \begin{array}{cc}q& , x=0\end{array}\\ \begin{array}{cc}\frac{\sqrt{x+x^2}-\sqrt{x}}{x^{3/2}}& , x>0\end{array}\end{array}\right. \end{gathered}$$ is continuous for all $x$ in $R,$ are

Define $F\left(x\right)$ as the product of two real functions $f_1\left(x\right)=x, x\in R,$ and $f_2\left(x\right)=\left\{\begin{array}{ccc}\sin \frac{1}{x},& \text{ If }& \mathrm{x}\ne 0\\ 0,& \text{ If }& x=0\end{array}\right.$ as follows

$F\left(x\right)=\left\{\begin{array}{ll}{f}_1\left(x\right)·f_2\left(x\right)& \text{ If } x\ne 0\\ 0,& \begin{array}{cc}\text{ If }& x=0\end{array}\end{array}\right.$

Statement -$1: F\left(x\right)$ is continuous on $R.$

Statement - $2: f_1\left(x\right)$ and $f_2\left(x\right)$ are continuous on $R.$

Consider the function, $f\left(x\right)=|x-2|+|x-5|, x\in R.$

Statement- $1:\hspace{0.17em}{f}^{\text{'}}\left(4\right)=0$

Statement- $2: f$ is continuous in $[2, 5],$ differentiable in $(2, 5)$ and $f\left(2\right)=f\left(5\right).$

Let $f,g:[-1,2]\to R$ be continuous functions which are twice differentiable on the interval $(-1,2)$. Let the values of $f$and $g$ at the points $-1,0$ and $2$ be as given in the following table:

In each of the intervals $(-1,0)$ and $(0,2)$, the function $(f-3g{)}^{\text{'}\text{'}}$ never vanishes. Then, the correct statement(s) is/are