Final answer:
The system has an unspecified number of critical points indicated by where the first derivative of potential energy is zero. Stability is determined by the second derivative; positive implies stability and negative indicates instability. Without the specific function, we cannot determine the exact number or largest x-coordinate of the critical points.
Step-by-step explanation:
From the information given, we are dealing with a system that has a point where the second derivative of potential energy is negative (indicating instability) and points where the second derivative is positive (indicating stability). Specifically, the critical points of a system represent the values where the first derivative is zero; these can be points of stable or unstable equilibrium.
The number of critical points can typically be determined by analyzing the potential energy curve of the system or by setting the derivative of the function representing the system's potential energy to zero and solving for the critical points. Regarding the stability of these points, if the second derivative is positive at a critical point, it's a stable equilibrium; if it's negative, it's an unstable equilibrium.
Without the specific function or potential energy curve, we cannot determine the exact number of critical points or the largest x-coordinate of these points. However, the system's linearization at the critical point with the largest x-coordinate will involve evaluating the second derivative at this point, and if it's positive, the behavior near this point will be stable.