Science:Math Exam Resources/Courses/MATH105/April 2011/Question 07
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Question 07 

Find numbers and such that the function has a critical point at Is this critical point a local maximum, a local minimum, or a saddle? 
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Hint 

What do partial derivatives at critical points look like? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The derivative of with respect to is
Also, the derivative of with respect to is
For a critical point at , we require that . Using the above two equations, this gives us the following system to solve for and :
Equation 2 tells us that . Substituting this value for into equation 1 tells us that . Therefore, we have that . This critical point is a saddle point. To show this, we solve for the second derivatives , and . First,
Therefore,
Second,
Therefore, Third,
Therefore,
We can classify the critical points by looking at the formula, If then the critical point is a saddle point. If then we look to the sign of . If then we have a local minimum and if then we have a local maximum. For our problem, Therefore since the critical point is a saddle point. Advanced: Using the second partial derivatives of a function we can define a general Hessian matrix as which for our specific example is
The eigenvalues of this matrix are given by the solving the formula for . That is,
Solving gives us the answer
Because one eigenvalue is positive and one eigenvalue is negative, we conclude that the Hessian matrix is indefinite, and thus the point is a saddle point. 