I looking for a function $y=f(x)$ (with minimum number of parameters $c_1,c_2,c_3,...$) satisfying all of the following properties:
- $f'(x)>0 \text{ } \forall \text{ } x \ge 0$,
- $\lim_{x \rightarrow\infty}f(x)$ exists,
- Depending upon the values of the parameters, the number of inflection points of $f$ can be either $0$ or $1$ (as we require). So for example, $$y=c_1+c_2e^{{c_3}x}$$ is not a valid form, because whatever values of the parameters $c_1,c_2,c_3$, we can not have $1$ inflection point if we require that.
The only form I get (by trying) is:
$$y=c_1+c_2e^{c_3(x^{c_4}+c_5)^{c_6}}$$
For example,
both blue curve and red curve have the same form, but we can fix the number of inflection points to be $0$ or $1$ is we wish by changing the parameters.
I found this: $$y=c_1+c_2e^{c_3(x^{c_4}+c_5)^{c_6}}$$ can you suggest me other?
Any help would be appreciated. Thanks!