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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 : enter image description here

Robert Howard
  • 2,753
J.A
  • 333

1 Answers1

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$f(x)=(x-2)^4+3(x-2)^3+2(x-2)^2+1$ seems to fulfill those criteria.

Expanding all those terms and simplifying, that equation becomes $$f(x)=x^4-5x^2+8x^2-4x+1$$

enter image description here

Robert Howard
  • 2,753
  • To shift the function $a$ units to the right, you replace $x$ in the equation by $(x-a)$. I saw from your illustration that the function you were looking for was most likely a fourth-degree polynomial (from the number of turning points), and then I played around with the coefficients in Desmos until I found a function that worked. – Robert Howard Nov 21 '18 at 16:41