Let $\{x_n\}$ be a real sequence defined by: $$ x_1=a \\ x_{n+1}=3x_n^3-7x_n^2+5x_n $$
For all $n=1,2,3...$ and $a$ is a real number. Find all $a$ such that $\{x_n\}$ has finite limit when $n\to +\infty$ and find the finite limit in that cases.
My thought: The answer is $a\in [0,\frac{4}{3}]$ we find root of $f(x)=x$