Proving that a 4th degree polynomial function has four fixed points and finding those fixed points

Question

So I have this function: $g(x)=-\mu^3x^4+2\mu^3x^3-\mu^2(\mu+1)x^2+\mu^2x$ where $\mu \in [0, 4]$ and $x\in[0, 1]$. Mu is a constant parameter. I have to prove that this function has 4 fixed points and I have to calculate them which means finding them without numerical methods. I have no idea how to proceed for both of these questions so I was hoping maybe someone here could guide me. So far I was thinking that since $g(x) = x$ I could subtract x on both sides and obtain a function that is equal to zero and then show that the function has 4 zeros because it is a 4th degree polynomial but I am not sure about that. As for the fixed points, other than x=0 I don't know how I can find them, because I don't know how to factor this polynomial or the one obtained by dividing by x and subtracting by 1. Also, I have to prove that $g(x) \in [0, 1]$ for the ranges I gave earlier for $x$ and $\mu$. I tried using the derivative to find the max value of the function but once again I am stuck with a third degree polynomial.

I would be very grateful if anyone could help or at least point me in the right direction, I am totally stuck on this problem. Thanks

Answer 1

For the mentioned range for $\mu$, it is not true that you always have four fixed points... For instance,

• If $\mu = 1$, you have a single fixed point
• If $\mu = 2$, you have two fixed points
• If $\mu = 4$, you have four fixed points

Maybe you want to rephrase the question?