Extreme Theorem

In calculus, the extreme value theorem states that if a real-valued function f is continuous in the closed and bounded interval [a,b ], then f must attain its maximum and minimum value, each at least once. That is, there exist numbers c and d in [a,b ] such that: If a function is continuous on a closed interval , then has both a maximum and a minimum on . If has an extremum on an open interval , then the extremum occurs at a critical point. This theorem is sometimes also called the Weierstrass extreme value theorem. The Extreme Value Theorem states that a continuous function from a compact set to the real numbers takes on minimal and maximal values on the compact set. If f is a continuous function and closed on the interval [a,b ], then f has both a minimum and a maximum. An important application of critical points is in determining possible maximum and minimum values of a function on certain intervals. The Extreme Value Theorem guarantees both a maximum and minimum value for a function under certain conditions. It states the following:

• If a function f(x)is continuous on a closed interval [a,b ], then f(x) has both a maximum and minimum value on [a,b ].

The procedure for applying the Extreme Value Theorem is to first establish that the function is continuous on the closed interval. The next step is to determine all critical points in the given interval and evaluate the function at these critical points and at the endpoints of the interval. The largest function value from the previous step is the maximum value, and the smallest function value is the minimum value of the function on the given interval.