I am looking for two positive functions f and g such that their sum h=f+g has minimum at 1/2 and maximum values at 0 and 1. The function h is strictly decreasing between 0 and 1/2 and strictly increasing from 1/2 to 1.

What candidate functions do exit?

I'm looking for specific functions f and g.

  • 1
    $\begingroup$ Are you somehow trying to qualify the set of all candidate function pairs? If not this is a bit of a boring question, I think. $\endgroup$
    – miradulo
    Jan 31, 2017 at 14:38
  • $\begingroup$ Find a function $f$ with a minimum at $\frac 12$, and take $g = 0$ $\endgroup$ Jan 31, 2017 at 14:39
  • $\begingroup$ @Omnomnomnom See the edit $\endgroup$ Jan 31, 2017 at 14:44

1 Answer 1


One can guess that Cosine function has this type of properties. So we can construct such function. as $f(x)=\frac{\cos(2\pi x)}{2}$ and take $g(x)=\frac{1}{2}$. But since you want positive functions so $f(x)=g(x)=\frac{\cos^2(\pi x)}{2}$ would be alright..

  • $\begingroup$ Nice, but I prefer non constant functions f and g. $\endgroup$ Jan 31, 2017 at 15:01
  • $\begingroup$ what about $f(x)=g(x)=\frac{cos^2(\pi x)}{2}$ $\endgroup$
    – Arun
    Jan 31, 2017 at 15:05
  • $\begingroup$ Can you plot it? $\endgroup$ Jan 31, 2017 at 15:06
  • $\begingroup$ you can check in this site. $\endgroup$
    – Arun
    Jan 31, 2017 at 15:08

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