Final answer:
The global extrema of a potential energy function can be determined by graphing the function, finding where it crosses zero, and computing the first and second derivatives to identify local maxima and minima. Also, examining the behavior at infinity confirms global behavior. This process is a common technique in physics to ensure that answers are physically plausible.
Step-by-step explanation:
Finding Global Extrema of a Potential Energy Function
To determine the global extreme values of a potential energy function, we follow a systematic approach. First, graphing the function is essential. The function described seems to have a double well shape with zeros where the potential energy U(x) equals zero. To find these zeros, we solve the equation U(x) = 0. Next, to locate the extrema, we compute the first derivative of U(x) and set it equal to zero to find the critical points. Critical points where the second derivative is positive indicate local minima (potential wells), and points where the second derivative is negative indicate local maxima.
Also, by evaluating the function's behavior as x approaches infinity, we can determine if there are any horizontal asymptotes that may represent global extrema. In physics, comparing the analytical results at extreme cases with physical expectations is a common technique for verifying reasonableness.
An example related to waves and deformations is mentioned, where the maximum deformation corresponds to the amplitude of the wave. In analyzing a graph, labeling the axes and identifying the scale with maximum values of x and y is important. For instance, plotting a function f(x) = 10 over the interval 0≤x≤20 would result in a horizontal line at y = 10, which represents a constant function within the given interval.