Let  $f\left(x\right)=\ln x,x>0$ and $g\left(x\right)=x,x>0$. Which of the following is/are true?

$\varphi \left(x\right)=\frac{{\left(f\left(x\right)\right)}^n}{g\left(x\right)}$has two critical points and local extreme values occur at both of these critical points b, for  $n\in N$

there exists k values $a_1,a_2,a_3,\mathrm{.....}{a}_k\in [1,k]$  such that $\frac{{\left(e\ln a_1\right)}^n}{a_1}+\frac{{\left(e\ln a_2\right)}^n}{a_2}+\frac{{\left(e\ln a_3\right)}^n}{a_3}+\mathrm{.......}+\frac{{\left(e\ln a_k\right)}^n}{a_k}\le 1$
  for  $n\in N$
 

$\phi \left(x\right)=\frac{{\left(f\left(x\right)\right)}^n}{g\left(x\right)}$   the both local maximum and local minimum points, for odd natural number n

${ }_{x\to \infty }^{lt}\frac{f\left(x\right)}{{\left(g\left(x\right)\right)}^{n/2}}=0$ , for  $n\in N$