Science:Math Exam Resources/Courses/MATH105/April 2014/Question 04 (b)
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Question 04 (b) 

Let . Find the maximum and minimum values of on 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 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. 
Hint 1 

Sketch the region and check the local max, local min, and points on the boundary. 
Hint 2 

On the boundary, transform the function to the one with only one variable and then differentiate it to find the minimum and maximum. 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We just need to check the local maximum, the local minimum and the points on the boundary. From question above, we know that it has only one local minimum and no local maximum. Since is not on , we only need to find the min and max on the boundary. On :
The critical points from above occur when which is at , neither of which are in . Thus, we just test the endpoints: and . These are two values we will consider later. On :
at and . As is not in we ignore it. We now compute and the value at the endpoints with and . From all of this, we find and . 