Extreme Value Theorem

The extreme value theorem states that if a real-valued function is continuous on a bounded closed interval , then attains both a maximum and a minimum on . It is also called the Weierstrass extreme value theorem. The image of the compact set under the continuous function is a compact set in , so it contains its supremum and infimum.

The closed interval method locates the global extrema by comparing the values of at the endpoints and all interior critical points, where or the derivative does not exist. Every interior extremum is a critical point, so the largest and smallest candidate values are the global maximum and global minimum, respectively.