how to find this function $f(x)$ satisfying these conditions?

Question

How to find a function $f(x)$ that satisfies:

1. $f(x)$ defines only on the positive axis of X;
2. when $x\to 0$, $f(x)\to +\infty$.
3. Exist a positive real number $k$, when $x\to k$, $f(x)\to 0$ and $f(x)=0$ for $x>k$.
4. $f'(x)<0$ for $x\in[0,k)$, and $f'(k)=0$.
5. $f''(x)>0$.

Thanks.

Answer 1

$$f:\mathbb{R_+} \to \mathbb{R}$$ $$ f(x)=\begin{cases} \cot^2(x) & 0<x<\dfrac{\pi}{2}\\ 0 & x\geq \dfrac{\pi}{2}. \end{cases} $$

Answer 2

There are so many examples so that it is difficult to chose. One example is $$ f(x)=\begin{cases} x+\frac{1}{x}-2 & 0<x<1\\ 0 & x\geq 1. \end{cases} $$ Here $k=1$ of course.