The function must be continuous on a closed interval. When this is the case, the Extreme Value Theorem guarantees that the function will attain both its absolute maximum value and its absolute minimum value either inside the interval at a critical number or at the endpoints of the interval.

Part 1:    • Maximum and Minimum Values of a Function  
Part 2:    • Extreme Value Theorem and Fermat's Th...  
Part 3:    • Finding the Critical Numbers or Criti...