Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
2
Academic year
2022
Sporkz
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48
Closed and bounded regions The mathematical process of optimization involves finding the maximum or minimumvalue of a function over a region.If we want to find the maximum of a function of x on an interval and the function isdifferentiable on this interval, we will have two cases to consider. The maximum canoccur at a critical point or it can occur at the end point. Remember, a critical pointmeans that the second derivative has a zero at that point. That's one thing that canhappen. And the other thing that can happen is that the maximum occurs at aboundary point. To illustrate these two scenarios, we can use graphs. For example 1, our functionlooks like this, and on the interval where it is positive, its maximum value is 0. This isa critical point because g'(x0)=0, where g'(x) is the first derivative of f(x). In example2, our function looks like this and it has one critical point; however, it is not on theboundary of the interval (it's in the interior). And in more complex situations, thepattern is the same. It's just more difficult to identify critical points in a more complexsystem.
We should say something about this region. A bounded region is one that includesall points within it and only those points, such as a disk or a solid square. Anunbounded region extends beyond its boundary, such as the whole right upperquadrant of the plane. When the region is bounded, it is an interval in one dimension, and the entirepositive number line or all positive numbers in two dimensions.The region bounded by x and y is the one where both x and y are positive. In onedimension, a bounded region is an interval. In an unbounded region, it would be likethe whole line or all of the positive axis. So these are the ones that are like intervals.
Understanding Bounded Regions in Mathematical Optimization
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