As a domain expert in mathematical analysis, I often delve into the intricacies of functions and their behavior. When we talk about endpoints and critical points, we're discussing the properties of functions within the context of calculus, particularly in relation to their derivatives and extrema. Step 1: English Answer In calculus, an endpoint is a point that marks the boundary of a closed interval. On the other hand, a critical point is a point at which the derivative of a function is either zero or undefined. These points are significant because they often correspond to local maxima or minima, or points of inflection where the function changes concavity. Now, can an endpoint be a critical point? The answer is nuanced and depends on the context of the function and the interval under consideration. 1. Local vs. Global Extrema: When we talk about extrema, we often distinguish between local and global extrema. A local extremum is a point where the function value is higher or lower than all nearby points, while a global extremum is the highest or lowest value of the function over an entire domain. An endpoint can be a local extremum if the function value at that point is greater than or less than the function values at all points in a neighborhood around it, even if that neighborhood does not include the entire domain. 2. Closed Intervals: For a function defined on a closed interval [a, b], the endpoints a and b are always considered in the analysis of extrema because they can indeed be points where the function attains its maximum or minimum values over the interval. If the function is continuous on [a, b], by the Extreme Value Theorem, it will attain both a maximum and a minimum value somewhere on the interval, which could be at the endpoints or at critical points within the interval. 3. Derivative Conditions: For a point to be considered a critical point in the traditional sense, the derivative at that point must be zero or undefined. However, endpoints of a closed interval do not have a derivative in the traditional sense because they are not interior points of the interval. Nonetheless, they can still be points where the function attains a local extremum. 4. Role of the Derivative: The derivative of a function at a point gives us information about the slope of the tangent line to the function at that point. At a critical point where the derivative is zero, the function changes from increasing to decreasing or vice versa. If the derivative is undefined, as it might be at a sharp corner or a cusp, the function may also exhibit an extremum. 5. End Behavior and Asymptotes: Sometimes, the behavior of a function at the endpoints can be influenced by the function's end behavior or asymptotes. For instance, if a function approaches a vertical asymptote as x approaches an endpoint, the endpoint itself may not be a critical point, but the behavior near the endpoint is significant. 6. Contextual Considerations: The determination of whether an endpoint is a critical point can also depend on the specific problem context. For example, in optimization problems, endpoints are often included in the search for optimal solutions. 7. Examples: To illustrate, consider a simple function like f(x) = x^2. On the closed interval [0, 3], the endpoints are 0 and 3. The function has a critical point at x = 0 (since the derivative f'(x) = 2x is zero there), which is also a local minimum. The point x = 3 is not a critical point in the traditional sense since the derivative does not exist there, but it is an endpoint where the function attains its maximum value on the interval. In conclusion, while endpoints are not typically critical points in the sense of having a derivative that is zero or undefined, they can still be points of extremum for a function on a closed interval. Whether or not an endpoint is considered a critical point depends on the broader definition one is using and the context of the problem at hand. **read more >>

But we want to define critical points to be all possible extrema. ... So for this function, extrema for a closed interval occur at the endpoints). classic critical points (interior points where the derivative is zero). Depending on other considerations, such a point may be a local extremum only, or nothing special.read more >>