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The function we are looking for is $f(x)$

We know, these values:

$f(0) \approx 0$

$f(1) = 1$

$f(2) \approx 0$

The limits are:

$\lim\limits_{x \to -\infty} f(x) = 0$

$\lim\limits_{x \to +\infty} f(x) = 0$

If $x < 1$ then $f(x)$ is strictly increasing, and if $x > 1$ then $f(x)$ is strictly decreasing

I have no idea, where to start.

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    $\begingroup$ Are there any more restriction on $f$? There can be a lot of such a function satisfying the given conditions. $\endgroup$ – BigbearZzz Nov 8 '15 at 13:58
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The first which came to my mind is a gaussian curve which has all the properties you want $$f(x)=e^{-k(x-1)^2}$$ Now, adjust the value of $k$ on what you want at $x=0$. Suppose that your goal is $f(0)=a$. Then $k=-\log(a)$.

But, as BigbearZzz commented, there could be many other functions.

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