Endpoints are really important when we're looking at special points in a function.

Many people might only pay attention to critical points, which are where the derivative equals zero, but ignoring the endpoints can make us miss important details about how the function behaves in a certain range.

When we find critical points, we're figuring out where the function changes its rate. These points show where the highest (local maxima) or lowest (local minima) values might be. But they don't tell us everything we need to know. Endpoints, which are the beginning and end of our interval, are really important too. They help us understand how the function acts overall.

For example, let’s think about a function that is continuous on a closed interval from $a$ to $b$. The local highest and lowest points can be found in two places:

1. At the critical points, where the derivative $f'(x) = 0$ or doesn’t exist.
2. At the endpoints $x = a$ and $x = b$.

To find the absolute highest or lowest value, we need to look at the function at both the critical points and the endpoints. The actual highest or lowest value can be at either of these spots, not just among the critical points.

If we forget to check the endpoints, we might miss the largest or smallest values the function can reach, which can lead to mistakes when studying how the function behaves. So, when searching for local extrema, we should always include the endpoints in our checks. They aren’t just the edges; they’re a key part of understanding the function.