Finding Absolute Maxima and Minima
In Calculus I, we first learned how to find and classify critical points, which allow us to find the location of
local maxima and minima. We also discussed the application of finding the absolute maximum and minimum
values of a function y“fpxqover a closed interval ra,bs. Recall that the absolute maximum and minimum
can only occur at either critical points or at the endpoints of the interval but nowhere else, assuming that
fpxqis continuous on ra, bs. This is known as the Extreme Value Theorem, and our goal is to now extend
these ideas to functions of two or more variables.
Once again, it should be clear that an absolute maximum or minimum value can occur at a critical point,
but what corresponds to the endpoints? First, absolute extrema are only guaranteed to exist over a set D
that is both closed and bounded. What does that mean? A set Dis called closed if it includes its boundary.
For example, the set defined by x2`y2ď1 is closed, but x2`y2ă1 is not (the latter set would have a
dashed circle for its boundary while the first would be solid). A set Dis called bounded if it doesn’t extend
infinitely in any direction. Mathematically, this is equivalent to being able to fit the set Dinside of a disk
with some finite radius.
We can now state the Extreme Value Theorem for a function of two variables. If fpx, yqis continuous on a
closed, bounded set Din the plane, then fwill attain an absolute maximum and minimum value at either
a critical point inside Dor on the boundary of the set D.
Example 1: Find the absolute maximum and minimum values of fpx, yq “ x2`y2`x2y`4 on the set D,
the rectangle given by ´1ďxď1, ´1ďyď1. Note: the critical points of fpx, yqare p0,0q,p?2,´1q, and
p´?2,´1q.
The critical points p?2,´1qand p´?2,´1qare not contained in the region D, so p0,0qis the only critical
point that we need to consider. fp0,0q “ 4, which we will use to determine the absolute minimum and
maximum values.
y
x
1´1
1
´1
Next, we need to determine the maximum and minimum values on each piece of the boundary curve, so
there are four possible line segments to consider:
If y“1, then fpx, 1q “ 2x2`5, which has a minimum on r´1,1sof fp0,1q “ 5 and maximum of fp˘1,1q “ 7.
If y“ ´1, then fpx, ´1q “ 5, so the value for any xin r´1,1sis 5.
If x“1, then fp1, yq “ y2`y`5, which has a minimum on r´1,1sof fp1,´1{2q “ 19{4 and maximum of
fp1,1q “ 7.
If x“ ´1, then fp´1, yq “ y2`y`5, which has a minimum on r´1,1sof fp´1,´1{2q “ 19{4 and maximum
of fp´1,1q “ 7.
So, comparing the maximum and minimum values at the critical point p0,0qand around the various boundary
curves, we have that the absolute maximum value of fpx,y qis 7 and the absolute minimum value is 4.
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