Science:Math Exam Resources/Courses/MATH105/April 2015/Question 03 (b)
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Question 03 (b) 

Find the maximum and minimum values of the function over the 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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

The maximum and the minimum values of can be attained only at critical points (i.e., where all partial derivatives of vanish) or on the boundary of . Compute the critical points of the function and the value the function takes at those points. Then, compare these to the extrema on the boundary, which were computed in Question 3(a), to determine the global maximum and minimum of over . 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The minimum or maximum of can only occur at its critical points (points where both are zero) or on the boundary of . We will find the critical points (the extrema on the boundary were found in Question 3(a)) and then compare all these points to determine the global maximum and minimum of on . Since there is a single critical point in where ; namely, . At this point, .
