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I was wondering if there is any systematic way to come up with a function with specific properties.

I want to have a concave function in domain of $[0, 1]$ where $f(0) = 0$ and $f(1) = 1$. In other words it is concave down, but it crosses $f(x) = x$ at $x = 0$ and $x = 1$. I also want to have a parameter that controls the concavity.

I was thinking of $-kx^2 + (k+1)x$ with trial and error.

In general, what is the best way to come up with a function having specific properties?

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If $f:[0,1] \to \mathbb R$ is twice differentiable, then we have:

$f$ is concave $ \iff f'' \le 0 $ on $[0,1]$.

Now let $f$ be of the form $f(x)=ax^2+bx+c$. Then it is easy to see, that $f$ is concave, $f(0)=0 $ and $f(1)=1 \iff a \le 0$ and $b=1-a.$

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