Science:Math Exam Resources/Courses/MATH200/December 2013/Question 03 (a)
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Question 03 (a) 

Find the minimium of the function subject to the constraint using the method of Lagrange multipliers. 
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 requires you to determine the critical points of a function F having the form where f(x,y,z) is the function that you are trying to optimize and your constraint is expressed as g(x,y,z) = 0. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Using Lagrange multipliers we define the function F where Determining the critical points of this function gives a set of points where the minima of f(x,y,z) occur subject to the given constraint. Computing the appropriate partial derivatives and equating them to zero gives: Solving (3) gives or . Considering the case first, we can try to solve for x using (1) giving us ..but this is clearly inconsistent. Thus while is a solution to (3), it gives inconsistent results for at least one other equation and hence is not a valid condition for us to consider. Moving onto the case z = 0, we can solve for in (1) and (2) giving us Solving for y in terms of x gives Plugging both the result above and z = 0 into (4) gives And hence and are two critical points. If we evaluate the value of f(x,y,z) at these critical points we obtain Thus, the minimum value of f subject to the given constraint is and it occurs at the point 