Extreme Value Theorem

The Extreme Value Theorem (EVT) is an important theorem in calculus that states that if a function is continuous on a closed interval, then the function attains both a maximum and minimum value on that interval. In other words, the function has at least one point in the interval where it reaches its highest value and at least one point where it reaches its lowest value.

The EVT is particularly useful in optimization problems, where the goal is to find the maximum or minimum value of a function within a given interval. The theorem relies on the properties of continuous functions and the concept of a closed interval. Applications of the EVT include problems in economics, engineering, and physics, where optimization is a key concern.