To solve this, let's analyze each statement based on the Stability Theorem for fixed points of a Discrete-Time Dynamical System (DTDS) with updating function $f$.

1. If $x$ is a fixed point of the DTDS and $|f'(x)| < 1$, then $x$ is a stable fixed point.

   • This is true because, for a fixed point $x$ to be stable, the derivative of $f$ at $x$ should have an absolute value less than 1, meaning small perturbations around $x$ decay over iterations.

2. If $x$ is a fixed point of the DTDS and $f'(x) < 1$, then $x$ is a stable fixed point.

   • This is incorrect. $f'(x) < 1$ does not necessarily imply stability unless $|f'(x)| < 1$. The derivative could be negative but with a magnitude greater than 1, leading to instability.

3. If $x$ is a fixed point of the DTDS and $f'(x) < 0$, then $x$ is a stable fixed point.

   • This is incorrect. A negative derivative does not ensure stability unless $|f'(x)| < 1$. A value like $f'(x) = -2$ would actually make the fixed point unstable.

4. If $f(x) = x$ and $f'(x) > 1$, then $x$ is an unstable equilibrium.

   • This is true. If $f(x) = x$ (meaning $x$ is a fixed point) and $f'(x) > 1$, perturbations will grow, indicating that $x$ is an unstable fixed point.

5. If $f(x) = x$ and $|f'(x)| > 1$, then $x$ is an unstable equilibrium.

   • This is also true. Whether $f'(x) > 1$ or $f'(x) < -1$, the magnitude of $f'(x)$ being greater than 1 implies instability due to growing perturbations.

6. If $f(x) = x$ and $|f'(x)| > 1$, then $x$ is a root of $f$.

   • This is incorrect. $f(x) = x$ means that $x$ is a fixed point, not necessarily a root of $f$ (a root of $f$ would require $f(x) = 0$, not $f(x) = x$).

Conclusion

The statements that are true are:

• Statement 1
• Statement 4
• Statement 5

Would you like further explanation on any of these concepts?

Here are five follow-up questions to expand on this topic:

1. What is the significance of the derivative's absolute value in determining stability?
2. How does the concept of a stable fixed point differ from an unstable fixed point?
3. What are the implications of a fixed point being an attractor in a DTDS?
4. Can you explain why a fixed point with $|f'(x)| = 1$ is considered a borderline case?
5. How would this analysis change if we were discussing a continuous-time dynamical system?

Tip: For stability analysis in discrete systems, always examine the magnitude of the derivative at fixed points; values less than 1 indicate stability, while values greater than 1 indicate instability.