Minimum distance between the curves $f\left(x\right)=e^x\&g\left(x\right)=\ln x$ is

Consider the following statements:

${\mathbf{S}}_{\mathbf{1}}$: The function $y=\frac{2x^2-1}{x^4}$ is neither increasing nor decreasing.

${\mathbf{S}}_{\mathbf{2}}$: If $f\left(x\right)$ is strictly increasing real function defined on $R$ and $C$ is a real constant, then number of Solutions of $f\left(x\right)=c$ is always equal to one.

${\mathbf{S}}_{\mathbf{3}}$: Let $f\left(x\right)=x; x\in (0,1), f\left(x\right)$ does not has any point of local maximal minima.

${\mathbf{S}}_{\mathbf{4}}$: $f\left(x\right)=\left\{x\right\}$ has maximum at $x=6$ (here $\{.\}$ denotes fractional part function). State, in order, whether $S_1, S_2, S_3, S_4$ are true or false.