Science:Math Exam Resources/Courses/MATH105/April 2013/Question 04 (b)
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Question 04 (b) 

Let . Classify each critical point you found as a local maximum, a local minimum, or a saddle point of . 
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 

To classify critical points of a multivariable function, we need to use the Hessian, the matrix of second partial derivatives. 
Hint 2 

Evaluate the Hessian at the critical points, take the determinant and see what the resulting value tells you about the type of critical point. 
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. From part a), the critical points are and the first partial derivatives are The second partial derivatives are given by To classify the critical points, we need to compute the Hessian matrix, , of the function : Evaluating at the critical point gives The determinant of at the point is equal to , which is less than zero. Hence the point is a saddle point. Evaluating at the critical point gives The determinant of at the point is equal to which is greater than zero and so is not a saddle point of and must be either a local max or min. Since is less than zero, the point is a local maximum of . Therefore, (x,y) = (0,0) is a saddle point and (x,y) = (1/2,1) is a local maximum. 