# How to parametrize a function such that it approaches $f(0)=0$ and $f(1)=1$ with different speed

I need a function (polynome) that values $0$ at $0$ and $1$ at $1$ and has these values as local maxima and minima. So far so easy the straight solution is:

$$f(x) = -x^4+2x^2$$

Now I want to parametrize the slope increase with which the function approaches $0$ and $1$ respectively (the pointedness or flatness of the maxima and minima). I further want the pointedness identical for $0$ and $1$. How can this be done?

Edit to make the purpose clearer:

I want a non linear function which is almost linear around 0.5 but then approaches 1(0) faster as x tends towards 1(0) respectively. The behavior should be the same for both edge cases.

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How do you measure the "pointedness"? Without knowing this, there's no way to make it the same at two places. – Gerry Myerson Dec 20 '12 at 4:48
@Gerry: My suspicion (and it is only a suspicion), is that it means the multiplicity of the zeros of $f'(x)$ at $x=0,1$ are the same. – Cameron Buie Dec 20 '12 at 4:54
@CameronBuie: That sounds promising. So I just need to parametrize $$f′(x)$$ and then take the integral and I should have my function – Martin Dec 20 '12 at 12:04
@GerryMyerson: Like Cameron said. I don't need any exact measure only "more" or "less" pointed. – Martin Dec 20 '12 at 12:05
Maybe you could just sketch what you want freehand, pointing out the characteristics of interest. I am not sure what you mean by linear around 0.5-should it be straight, which is what linear means to me, or should the slope be small, which seems more in the spirit of the rest of the question? Ignoring the local max/min for a minute, is $f(x)=\frac 12+4(x-\frac 12)^3$ something like what you want? – Ross Millikan Dec 20 '12 at 14:35