Final answer:
To find the critical points of the differential equation dy/dx = (y - 1)^4, we set the derivative to zero, leading to the critical point y=1.
Step-by-step explanation:
The question involves analyzing the critical points of an autonomous first-order differential equation given by dy/dx = (y - 1)^4. To find the critical points of the differential equation, we set the derivative equal to zero. This happens only when y-1=0.
Thus, the critical point is y=1. At this point, the slope of the solution curves is zero, indicating we have a horizontal tangent. For the phase portrait, we observe the behavior of the solutions around the critical point. When y < 1, the derivative is positive, and when y > 1, the derivative remains positive.
This implies that the solutions will be increasing on both sides of the critical point; however, the increase is less steep as it approaches y=1 due to the fourth power in the derivative. The phase portrait would show a stable equilibrium at y=1 with arrows pointing towards the critical point.