Final answer:
To find the value of r at which the two-cycle is superstable, we solve equation f(f(p)) = p and f(f(q)) = q simultaneously. The values of p and q depend on the specific function and the value of r. There is no general rule to determine if one of them is a special value.
Step-by-step explanation:
In mathematics, a two-cycle refers to a cycle of length 2 in a function. To find the value of r at which the two-cycle is superstable, we need to consider the equation f(f(x)) = x, where f(x) represents the function. When the two-cycle is superstable, it means that the two points in the cycle converge to a stable fixed point under repeated iterations of the function.
Let's say the two points in the cycle are p and q. To find the value of r, we need to solve the equation f(f(p)) = p and f(f(q)) = q simultaneously. The values of p and q at this point depend on the specific function f(x) and the value of r. There is no general rule for determining whether any of the values are special in terms of the function f(x). It would depend on the specific function and the properties of the two-cycle.