Let $f:(0,\infty )\to R$ be a differentiable function satisfying the equation $2f\left(x\right)=f\left(xy\right)+f\left(\frac{x}{y}\right);\forall x>0,y>0$.  If $f\left(1\right)=0$ and $f^{\text{'}}\left(1\right)=1.$ Then which of following is/are true 

$f\left(x\right)=ex$ has no real solutions

$\underset{x\to 0}{\lim }\left[\frac{f(x+1)}{x}\right]=0,$ where [k] denotes greatest integer  function less than or equal to k

The equation $2e.f\left(x\right)=x$ has exactly one real solution

$f\left(x\right)$ has no local maxima and no local minima