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I need to find a function that has 4 fixed points, and all of them are unstable. I don't know how to proceed in this kind of problem, but i know how to find the fixed points in a function, and i know how to determine if the fixed points are unstable or stable, but i never worked constructing a function with $n-$fixed points. I think there is a general way to make those functions, but i don't know it.

Any hints?

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  • $\begingroup$ Dose it have to be a continuous function? $\endgroup$ – Q the Platypus Apr 29 at 3:39
  • $\begingroup$ @QthePlatypus isn't neccesary, but if it continuous i think it's harder to find. I'm not sure. $\endgroup$ – Rodrigo Pizarro Apr 29 at 3:40
  • $\begingroup$ A trivial solution would be to define your function pair wise so f(1) = 1 f(2) = 2 etc and f(n) = n^2 + 6 otherwise. $\endgroup$ – Q the Platypus Apr 29 at 3:45
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Hint: Try $f$ of the form $f(x) = x + c (x-x_1)(x-x_2)(x-x_3)(x-x_4)$. Now what do you need to be sure that $x_1, \ldots, x_4$ are unstable?

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  • $\begingroup$ I can take the derivative, but i don't see how to continue. $\endgroup$ – Rodrigo Pizarro Apr 29 at 4:46
  • $\begingroup$ How about, say, $x_1 = 1$, $x_2 = 2$, $x_3 = 3$, $x_4 = 4$? Can you find a $c$ that works? $\endgroup$ – Robert Israel Apr 29 at 13:58

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