Science:Math Exam Resources/Courses/MATH105/April 2014/Question 03 (a)
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Question 03 (a) 

Use the method of Lagrange multipliers to find the maximum and minimum values of on the circle . A solution that does not use the method of Lagrange multipliers will receive no credit, even if the answer is correct. 
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 method of Lagrange multipliers states that the extreme values of the objective function subject to the constraint will occur at the solutions to the system: . 
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 object function is and the constraint is , i. e. . By the method Lagrange multipliers, set which tells us . Looking at the vector equation in components, we have that Provided we don't divide by zero (so that ), we can divide (1) by (2) to yield where the first implication came by cross multiplying. If then from the constraint we must have and . Evaluating and . We still need to consider the possibility that . If then (2) reads . If then (1) tells us that so that . However, does not satisfy so this is not a valid solution to the Lagrange system. Overall have found that the maximum value is 180 and the minimum value is 80. 