Using the extreme value theorem to find absolute maxima and minima
Steps to solve extreme value theorem problems
In order to find global extrema of the function which is defined for a specific set of points, follow these steps:
Find first-order partial derivatives of the function
Use the first-order partial derivatives to find critical points, and eliminate any critical points that lie outside the set
Plug the remaining critical points into the original function to find the value of the function at those points
Find single-variable equations for the line segments that define the edges of the set
Take first-order partial derivatives of these line-segment equations with respect to the changing variable
Use the first-order partial derivatives of the line-segment equations to find critical points of each line segment
Plug the critical points from Step 6 into the original function to find the value of function at those points
Identify "corners" of the set and treat the coordinate points of the corners as critical points, plugging them into the original function to find the value of the function at those points
Compare the values of all the critical points (inside the set, along the edges, at the corners) to find the maximum and minimum values that the function attains in the set