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Absolute Minimums
and Maximums
In this section we are going to extend the work from the previous section
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In this section we are want to optimize a function, that is identify the
absolute minimum and/or the absolute maximum of the function, on a given region in 2
...
In order to optimize a function in a region we are going to need to get a couple of definitions out
of the way and a fact
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Definitions
2
1
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A region is called open if it
doesn’t include any of its boundary points
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A region in 2 is called bounded if it can be completely contained in a disk
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Let’s think a little more about the definition of closed
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Just what does this mean? Let’s think of a rectangle
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Open
Closed
−5 < x < 3
−5 ≤ x ≤ 3
1< y < 6
1≤ y ≤ 6
In this first case we don’t allow the ranges to include the endpoints (i
...
we aren’t including the
edges of the rectangle) and so we aren’t allowing the region to include any points on the edge of
the rectangle
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In the second case we are allowing the region to contain points on the edges and so will contain
its entire boundary and hence will be close
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Extreme Value Theorem
If f ( x, y ) is continuous in some closed, bounded set D in
( x1 , y1 )
2
then there are points in D,
and ( x2 , y2 ) so that f ( x1 , y1 ) is the absolute maximum and f ( x2 , y2 ) is the absolute
minimum of the function in D
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It only tells us that they will exist
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The basic process for finding absolute maximums is pretty much identical to the process that we
used in Calculus I when we looked at finding absolute extrema of functions of single variables
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Here is the process
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Find all the critical points of the function that lie in the region D and determine the function
value at each of these points
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Find all extrema of the function on the boundary
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3
...
The main difference between this process and the process that we used in Calculus I is that the
“boundary” in Calculus I was just two points and so there really wasn’t a lot to do in the second
step
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Let’s take a look at an example or two
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Solution
Let’s first get a quick picture of the rectangle for reference purposes
...
x = 1, − 1 ≤ y ≤ 1
x = −1, − 1 ≤ y ≤ 1
y = 1, − 1 ≤ x ≤ 1
right side :
left side :
upper side :
lower side :
y = −1, − 1 ≤ x ≤ 1
These will be important in the second step of our process