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I am looking for functions, $f(x)$ with the following properties:

  • $f>0, f’(x)>0, f’’(x)\leq 0$
  • f continuous on its domain $[a,d]$ where $0\leq a \leq d]$
  • $f’$ is constant or near constant on $[a,b]\cup[c,d]$ and gets much smaller somewhere on $[b,c]$

Are there some general forms for functions that would satisfy this?

Or is my best bet just to paste together functions like $f(x)=x$ On $[0,1]$, $2\sqrt{x}-1 $ on $[1,4]$ and $f(x)=x/4+2$ on $[4,10]$

The downside to paste-ing functions together is the 2nd derivative probably not being continuous

If pasteing functions together is the way to go, does anyone know some functions that are easy to manipulate how fast the first derivative is decreasing?

Thanks

Valent
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user106860
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    Check this question (also I have seen similar questions around): https://math.stackexchange.com/questions/3643915/if-fx0-and-fx-leq-0-for-x0-show-that-fx-geq-0-for-x0 – Valent Feb 18 '21 at 04:49

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