Equilibrium points of x-x^2-1/12 and sketch phase diagrams

Here is the phase diagram for the function \( f(x) = x - x^2 - \frac{1}{12} \). It shows the equilibrium points and the direction of the flow, indicating the stability and instability of the regions around the equilibria.

Would you like a more detailed explanation of how the phase diagram is interpreted?

Here are 5 related questions:
1. How do we determine the stability of equilibrium points from the phase diagram?
2. Can you explain the significance of the direction of arrows in phase diagrams?
3. How can we analyze higher-order stability (e.g., using the second derivative)?
4. What is the physical interpretation of equilibrium in dynamical systems?
5. How do different initial conditions affect the trajectory of a system?

**Tip:** For phase diagrams, identifying changes in stability helps predict system behavior under perturbations.

Explore how to find the equilibrium points of the function x - x^2 - 1/12 and learn to sketch the corresponding phase diagram. This tutorial covers key concepts such as stability analysis and phase diagrams, aimed at undergraduate students studying differential equations.

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