Science:Math Exam Resources/Courses/MATH105/April 2010/Question 01 (c)
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Question 01 (c)
Find all point(s) (x,y) where
may have a relative maximum or minimum.
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How do we define critical points for functions of two variables?
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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 .