In mathematical analysis, critical points are the points where a function’s derivative is either zero or undefined. However, when considering functions on a closed interval, it’s crucial to evaluate the role of the endpoints as well. In this context, critical points may or may not include endpoints.
Typically, when analyzing a function on a closed interval [a, b], we look for potential maximum and minimum values at:
- The critical points within the interval,
- The endpoints of the interval, a and b.
Thus, while critical points usually refer to the points where the derivative is zero or undefined, in the evaluation of extrema on a closed interval, it is standard practice to include the endpoints. This is because the maximum or minimum value of a function on that interval can occur at either critical points or the endpoints.
To summarize, in the context of extrema on a closed interval, you should consider both critical points and endpoints when determining the overall behavior of the function. Critical points do not inherently include endpoints, but for practical purposes in optimization, both sets of points must be examined.