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CRGreathouse

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## Main Question or Discussion Point

I'm almost too embarrassed to post, but I thought someone might have insight that could help me here. I'm trying to fit an equation to data, but I'm just not sure what sort of equation I should use. I have some requirements on the form of the equation, and then I have points (yet to be determined, actually).

[tex]f'(x)>0\,\,\forall x>0[/tex] (cost is increasing)

[tex]f''(x)<0\,\,\forall x>0[/tex] (essentially, average cost is decreasing)

[tex]\lim_{x\rightarrow\infty} f(x)=k[/tex] (price is bounded above in this fashion)

It would make the most sense if the origin was included, but this can be one of the data points.

So before I even think about least-squares vs. minimal points on the curve, I wanted to consider different forms that could make sense. The first that comes to mind was the elementary

[tex]y=a-\frac{ac}{x+c}[/tex]

but this has some instabilities and oddities. I guess I'm just looking for thoughts on widely-used models that fit my criteria or could be modified to fit them. In the perfect case I'd have a smooth equation that was nearly linear for small x and essentially hyperbolic for large x.

Any suggestions would be welcomed.

[tex]f'(x)>0\,\,\forall x>0[/tex] (cost is increasing)

[tex]f''(x)<0\,\,\forall x>0[/tex] (essentially, average cost is decreasing)

[tex]\lim_{x\rightarrow\infty} f(x)=k[/tex] (price is bounded above in this fashion)

It would make the most sense if the origin was included, but this can be one of the data points.

So before I even think about least-squares vs. minimal points on the curve, I wanted to consider different forms that could make sense. The first that comes to mind was the elementary

[tex]y=a-\frac{ac}{x+c}[/tex]

but this has some instabilities and oddities. I guess I'm just looking for thoughts on widely-used models that fit my criteria or could be modified to fit them. In the perfect case I'd have a smooth equation that was nearly linear for small x and essentially hyperbolic for large x.

Any suggestions would be welcomed.