find a such function $f(x)$

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. For a positive real number $k$, when $x\to k$, $f(x)\to 0$.
4. for $x\geq k$, $f(x)=0$.
5. $f'(x)<0$ for all $x\leq k$.
6. $f''(x)>0$ for all $x\leq k$.

I think it has a form of $f(x)=\frac{1}{x}$ or something else, but I dont know how to drag the $\infty$ to $k$. Thanks.

Answer 1

This one is even smooth: $$f(x)=\begin{cases}0&\text{if $x\ge k$}\\ \frac1xe^{-\frac{1}{k-x}}&\text{if $0<x<k$}\end{cases} $$

Answer 2

You can use $$f(x)=\begin {cases} \frac 1x-\frac 1k & x \le k \\0 & x \gt k \end {cases}$$