Absolute and Local Extreme Values

The Extreme Value Theorem guarantees the existence of both an absolute maximum and an absolute minimum for any continuous function defined over a closed interval.

These values can be found where the derivative is zero, undefined, or at the interval’s endpoints.

Consider how temperature changes over a day, modeled as a continuous function over a closed interval. The highest and lowest temperatures are called the absolute maximum and the absolute minimum, or extreme values. These values help summarize the day’s weather.

Over a shorter time span, a temperature higher or lower than nearby values is a local maximum or local minimum.

Now consider another function that describes how voltage changes in an alternating current. It can be modeled as a sine function over time. The voltage rises and falls in a repeating pattern, creating multiple points where the slope is zero. These points are local maxima and minima. Over a closed interval, such as one cycle, the highest and lowest of these values are the absolute extrema. These extrema define the safe operating range for circuits.