Final answer:
To find the x-coordinates of critical points of a function, calculate the first derivative and set it to zero. The second derivative test determines the nature of critical points; a negative value suggests a relative maximum (unstable equilibria), and a positive value indicates a relative minimum (stable equilibria).
Step-by-step explanation:
Finding Critical Points and Determining Their Nature
To find the x-coordinates of all critical points of a function, you first need to find the function's first derivative and set it equal to zero. This calculation will reveal the x-values where the slope of the tangent line to the function's graph is zero, indicating potential critical points. To determine the nature of these critical points, you apply the second derivative test.
If the second derivative of the function is negative at a critical point x, this implies that the function is concave down at that point, indicating a relative maximum. On the other hand, if the second derivative is positive at a critical point x, this implies that the function is concave up at that point, suggesting a relative minimum. These points also correspond to unstable and stable equilibria, respectively.
When analyzing potential energy as a function of position, a graph that resembles a double potential well typically indicates stable equilibria at the wells' bottoms (where the second derivative is positive) and unstable equilibria at the top (where the second derivative is negative).