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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.

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    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. – 2017-01-31
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    Find a function $f$ with a minimum at $\frac 12$, and take $g = 0$ – 2017-01-31
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    @Omnomnomnom See the edit – 2017-01-31

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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..

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    Nice, but I prefer non constant functions f and g. – 2017-01-31
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    what about $f(x)=g(x)=\frac{cos^2(\pi x)}{2}$ – 2017-01-31
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    Can you plot it? – 2017-01-31
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    you can check in this [site.](https://www.desmos.com/calculator) – 2017-01-31