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Find the absolute maximum and minimum values of the function on the rectangular region .
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.
First, find the critical points of (where ) in the interior of the domain, then check the boundary. The extrema must be at one of the critical points, or on the boundary.
To handle the boundary case, break up the boundary into 4 cases: , , , and . For the first line, you get a new function , which you can then optimize subject to the constraint . The other lines are similar.
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.