Science:Math Exam Resources/Courses/MATH105/April 2010/Question 01 (c)
Question 01 (c)
Find all point(s) (x,y) where
may have a relative maximum or minimum.
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!
How do we define critical points for functions of two variables?
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
For to have a local maximum or minimum at a point , that point needs to be a critical point, that is
For the function
we have the partial derivatives
To look for critical points, we need to find points which yield a zero in both partial derivatives. Looking in particular at the partial derivative with respect to y we observe that if
Taking this value of and substituting it into the equation for the partial derivative with respect to yields
We find only one critical point,
Since in order to be a relative maximum or minimum, the point must be a critical point, we have found that the only point that may be a maximum or minimum is (2,5/12).
Continuing the Problem:
If we assume the second derivative test is applicable (sometimes it is inconclusive), we need
to have a relative maximum or minimum.
If we compute the second derivatives, we find
Since this is a negative number, this critical point is a saddle point and so the conclusion is that the critical point is not a relative maximum or minimum and since there is only one critical point, there are no relative maxima/minima for .