Questions on the tent map, $T_c (x)$ Suppose we have the tent map, defined by $$T_c(x) = \begin{cases} cx & 0\leq x \leq 1/2\\ c-cx& 1/2 \leq x \leq 1. \end{cases}$$
What are the prime period 3 orbits for this map? I have seen them for the tent map when $c=2$, but I cannot find them in terms of $c$ for the life of me.

The line segment of $T_c^3$ have equations and fixed points
\begin{array}{c|rl|c}
0&c(c(c(x)))&=c^3x&\\
1&c(1-c(c(x)))&=c-c^3x&\dfrac{c}{1+c^3}\\
2&c(c(1-c(x)))&=c^2-c^3x&\dfrac{c^2}{1+c^3}\\
3&c(1-c(1-c(x)))&=c-c^2+c^3x&\dfrac{c}{1+c+c^2}\\
4&c(c(c(1-x)))&=c^3-x^3&\dfrac{c^3}{1+c^3}\\
5&c(1-c(c(1-x)))&=c-c^3+c^3x&\dfrac{c+c^2}{1+c+c^2}\\
6&c(c(1-c(1-x)))&=c^2-c^3+c^3&\dfrac{c^2}{1+c+c^2}\\
7&c(1-c(1-c(1-c)))&=c-c^2+c^3-c^3x&\dfrac{c}{1+c}
\end{array}
where one can recognize the two 3 cycles
$$
\dfrac{c}{1+c^3}\to\dfrac{c^2}{1+c^3}\to\dfrac{c^3}{1+c^3}
$$
and 
$$
\dfrac{c}{1+c+c^2}\to\dfrac{c^2}{1+c+c^2}\to\dfrac{c+c^2}{1+c+c^2}
$$
These exist for
$$
\dfrac{c}{1+c+c^2}\le \frac12\le\dfrac{c^2}{1+c+c^2}\\\iff\\
2c\le1+c+c^2\le 2c^2\\\iff\\
-c^2+c-1\le 0 \le c^2-c-1
$$
which is true for $c\ge\frac{1+\sqrt5}2$, see also Prove that the critical value for the appearance of period 3-orbits in the tent map is $\frac{1+\sqrt5}{2}$.
