Science:Math Exam Resources/Courses/MATH105/April 2011/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?
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!
What do partial derivatives at critical points look like?
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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 .
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.
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 .
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.