I'm looking for an analytical expression of a function that would have a certain kind of behavior:
$\lim_\limits{x \rightarrow + \infty}f(x)=+\infty$ (or at least $\lim_\limits{x \rightarrow B'>B}f(x)=+ \infty $)
$\lim_\limits{x \rightarrow - \infty}f(x)=+\infty$ (or at least $\lim_\limits{x \rightarrow A'<0}f(x)= + \infty$)
Two inflection points $a,b$ with $a<b,f''(a)=f''(b)=0$
Two points with the same derivative $A,B$ with $A<a<b<B$, $f'(A)=f'(B) \geq 0$
$A,B>0$
Like here :
$f(x)=(x-2)^4+3(x-2)^3+2(x-2)^2+1$ seems to fulfill those criteria.
Expanding all those terms and simplifying, that equation becomes $$f(x)=x^4-5x^2+8x^2-4x+1$$