As an exercise, I need to find a $C^2$ function $f:\mathbb{R}^*\to\mathbb{R}$ such that $\eta>0$ exists such that $$ 1.\quad |f^{\prime}(x)|\le \eta\quad \mbox{ as } x\to +\infty;$$ $$ 2.\quad f^{\prime}(x)\ge \frac{1}{x^2}\quad \mbox{ as } x\to 0;$$ $$3. \quad a\in\mathbb{R}^* \mbox{ exists such that } f^{\prime}(a)=a \mbox{ (i.e. $f^{\prime}(a)$ has a fixed point)};$$ $$4.\quad f^{\prime\prime} (a)<-1. $$
I am thinking about the function $f(x) =-\frac{1}{x}$; in fact $\lim_{x\to+\infty} f(x)=0$ and $2.$ is satisfied with the equality. Moreover, $f^{\prime}$ has the fixed point $a=1$ and $f^{\prime\prime}(x)=\frac{-2}{x^3}$ so that $4.$ holds true.
Could someone please tell me if my computations hold true? If possible, could someone please give me another (or more than another one) example of function satisfying these assumptions.
Thank you in advance!