Final Answer:
The critical points are x = 11. The stability analysis indicates that x = 11 is a stable critical point. The explicit solution for x(t) in terms of t is x(t) = 11 + Ce^(t), where C is a constant determined by the initial condition x(0) = x₀. Visual inspection using slope fields confirms the stability of the critical point at x = 11.
Step-by-step explanation:
To find the critical points, set f(x) = (x - 11)^2 = 0. The only solution is x = 11. To determine stability, analyze the sign of f(x) around x = 11. For values of x less than 11, f(x) is positive, indicating the system moves away from x = 11. For x greater than 11, f(x) is negative, suggesting the system moves towards x = 11. Hence, x = 11 is a stable critical point.
The explicit solution for x(t) is found by solving the differential equation. Integrating dt/dx = (x - 11)^2 with respect to x gives x(t) = 11 + Ce^(t). The constant C is determined by the initial condition x(0) = x₀. Visually verifying stability using slope fields involves plotting tangent lines at different points. For the given differential equation, solution curves approaching x = 11 from either side indicate its stability.
By incorporating an initial condition like x(0) = x₀, the specific constant C can be determined. Plotting solution curves in a slope field allows visualization of how trajectories move concerning the critical point x = 11, affirming its stability when trajectories converge towards it.