Let $f\left(x\right)=x-\frac{1}{x}$, then which one of the following statement(s) is/are true?

If $f\left(x\right)=\left\{\begin{array}{ll}-e^{-x}+k& , x\le 0\\ e^x+1& , 0<x<1\\ e{x}^2+\lambda & , x\ge 1\end{array}\right.$is one-one and monotonically increasing $\forall x\in R,$ then

Let $h$ be a twice continuously differentiable positive function on an open interval $H.$ Let $g\left(x\right)={\log }_e\left(h\left(x\right)\right)$ for each $x\in H$. Suppose ${\left(h^{\text{'}}\left(x\right)\right)}^2>h^{\text{'}\text{'}}\left(x\right)h\left(x\right)$ for each $x\in H$.

Then, we can conclude that: