In optimization, what do you typically evaluate at critical points?

In optimization, evaluating the values of the function at critical points is crucial for determining extrema (maximum and minimum values). Critical points are where the derivative of the function is either zero or undefined. At these points, you can assess whether the function reaches a local maximum, local minimum, or neither.

By examining the function's values at these critical points, you can compare them and identify which are the highest or lowest values within the considered interval. This step effectively guides the optimization process, allowing you to make informed conclusions regarding the function's behavior in terms of extremum points, which are pivotal in many applications, such as economics, engineering, and resource management.

While evaluating the slope can indicate where potential extrema may occur, and checking the continuity of the function ensures no abrupt changes exist in its domain, the primary goal of optimization is to find the best values of the function, making the evaluation of its values at critical points the most relevant choice in this context. The domain often ensures that you are working within valid limits but does not directly involve the process of finding extrema.