Science:Math Exam Resources/Courses/MATH105/April 2018/Question 03
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Question 03 

Use the method of Lagrange multipliers to find the absolute extrema of the function over the boundary of the ellipse . Give the maximum and minimum values ( values), as well as all locations where they occur ( and values). This question evaluates your proficiency with the method of Lagrange multipliers, so a solution not using this method will receive no credit. 
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Hint 

By the method of Lagrange multipliers, find three equations that a local extremum for the function on the ellipse should satisfy. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let the constraint function . Note that the objective function is given by . By the method of Lagrange multipliers, if is a local extremum of the function , which can be a candidate for the absolute extrema, on the ellipse, then it satisfies
for some .
Since we have
a local extremum on the ellipse should satisfy
The second equation implies that , so either or should hold. In the case of , we get from the third equation. Then, from the first equation, we have , which is equivalent to and . Therefore, possible local extrema for the function on the ellipse are and . On the other hand, if , then the first equation gives , so that . Then, for some the third equation can be satisfiedwe don't need its value. Therefore, in this case, we obtain possible local extrema and . Finally, we compare the function value of at theses candidates, to find the absolute extrema. Since
on the ellipse, the function has its absolute maximum value at and while it has its absolute minimum value at .
Answer: 