When looking for local maxima and minima
Set
and solve simultaneously for
x
and
y
•
These points are called the
critical points
(CPs)
•
We test them to see whether or not they are local minima, maxima or saddle points
•
(at the CPs found above)
When looking for an absolute maximum or minimum
Consider a surface from the side,
over some bounded domain D (in blue).
Finding critical points and using the second derivative test, we will be able to find the two local maxima
Using the second derivative test
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But we can see that the function has even higher values in the domain
These are not covered by the critical points, since they are not
turning points or saddle points (remember
critical points show where the surface has slope 0, and above at the maximum value on the right it does
not have slope 0, so the CPs will not include this point).
Therefore after evaluating CPs, we must also evaluate the function along the boundary of its domain.
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 Spring '14
 Critical Point

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