5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points

The Extreme Value Theorem says that if a function is continuous on a closed interval [a,b], then it must attain both an absolute maximum value and an absolute minimum value somewhere on that interval. These are also called global extrema. A local maximum is a value that is greater than nearby function values, while a local minimum is a value that is less than nearby function values. A point where a function may have a local maximum or minimum often occurs at a critical point, which is a point in the domain where f'(x)=0 or where f'(x) does not exist. This lesson is about distinguishing between global and local behavior and identifying the points where important changes can happen.