For every twice differentiable function  $f:R\to [-3,3]$  with  ${\left(f\left(0\right)\right)}^2+{\left(f^{/}\left(0\right)\right)}^2=100$ , which of the following statement(s) is (are) TRUE?

Applying LMVT on  $[-6,0]$
$f^{/}\left(x_0\right)=\frac{f\left(0\right)-f(-6)}{0-(-6)}=\frac{f\left(0\right)-f(-6)}{6}$  for  $a_0\in (-6,0)$
 $\left|f^{/}\left(x_0\right)\right|=\left|\frac{f\left(0\right)-f(-6)}{6}\right|\le \frac{\left|f\left(0\right)\right|+\left|f(-6)\right|}{6}$
   $\le \frac{3+3}{6}=1$
If f(x) is periodic then  $\underset{x\to \infty }{\lim }f\left(x\right)\ne 1$
Similarly by applying LMVT on [0, 6]
$\left|f^{/}\left(x_1\right)\right|\le 1$   for   $x_1\in (0,6)$
Consider  $g\left(x_0\right)={\left(f\left(x_0\right)\right)}^2+{\left(f^{/}\left(x_0\right)\right)}^2\le 10$
 $g\left(x_1\right)={\left(f\left(x_1\right)\right)}^2+{\left(f^{/}\left(x_1\right)\right)}^2\le 10$
Let  $\alpha$  be the local maximum of  $g\left(x\right)\Rightarrow g^{/}\left(\alpha \right)=0,g^{//}\left(\alpha \right)\le 0$
 $\Rightarrow 2f\left(\alpha \right)f^{/}\left(\alpha \right)+2f^{/}\left(\alpha \right)f^{//}\left(\alpha \right)=0$
 $\Rightarrow f^{/}\left(\alpha \right)(f\left(\alpha \right)+f^{//}\left(\alpha \right))=0\Rightarrow f\left(\alpha \right)+f^{//}\left(\alpha \right)=0$
If   $f^{/}\left(\alpha \right)=0$ then  $g\left(\alpha \right)$  is not local maximum of g(x) at local maximum  
$f^{/}\left(\alpha \right)\ne 0$