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Lecture

# Absolute Extrema.pdf

7 Pages
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School
University of Toronto St. George
Department
Mathematics
Course
MAT133Y1
Professor
Abe Igelfeld
Semester
Fall

Description
5. ABSOLUTE EXTREMA De▯nition, Existence & Calculation We assume that the de▯nition of function is known and proceed to de▯ne \absolute mini- mum". We also assume that the student is familiar with the terms \domain" and \range" of a function. De▯nition. A real valued function on a set S (f : S ! R , where R denotes the real numbers) has an absolute minimum at the point a 2 S if f(a) ▯ f(x) for all x 2 S . Similarly, f has an absolute maximum at b 2 S if f(x) ▯ f(b) for all x 2 S . If we want to refer to one of absolute maximum or minimum without specifying which one we call it an absolute extremum. It is important to note that an absolute maximum requires that the largest value of f on the set is actually assumed at some point and that an absolute minimum requires that the smallest value of the function is actually assumed at some point. Examples: The following examples are chosen to illustrate a range of manifestations of the concepts of absolute maximum and minimum. (1) Let S be the set of students registered in MAT 133 on a given date. Let h(x) be the height of student x in cm . If a s the shortest student (or one of them, if there are several students who qualify for the shortest height) and b is the tallest student (or one of them if several students qualify for being tallest), then a and b are the absolute minimum and maximum points of the function h . (2) S = R;f(x) = x . f increases from ▯1 to 1 and has no absolute extrema. (3) S = R;f(x) = 1 . Every point is an absolute maximum and an absolute minimum. (4) S = [0;1];f(x) = x . f increases from 0 to 1 and has absolute minimum 0 at x = 0 and absolute maximum 1 at x = 1 . 3 2 2 2 (5) S = R ;f(a;b;c) = a + b + c . f has absolute minimum 0 at the origin, no absolute maximum. (6) In this example S is the set of real numbers R . Whenever S is a subset of R , graphs are very useful to help understand what is going on. Accordingly, let f : R ! R and 4 2 f(x) = x ▯ 2x . The graph looks like 1 y x It is evident from the graph that the lowest points occur at (-1,-1) and (1,-1). Consequently the points +1 and ▯1 are absolute minima of f . There is no highest point on the graph (it rises to 1 ) so there is no absolute maximum. (7) In this example S is a subset of the plane which has been ▯tted with the usual coordinate system. In such examples a specialized graphical representation is often used for real functions. Let 2 2 2 S = f(u;v) ▯ R ju + v ▯ 1g; f(u;v) = u + v . Here f has a single absolute maximum p p p p at (1 2;1= 2) and a single absolute minimum at (▯1= 2;▯1= 2) . v (1/ 2, 1 / 2) 1 1 u u + v = 2 (-1 / 2, -1 2) u + v = 1 u + v = 0 u + v = - 2 (8) Our ▯nal example is simple but important. It shows that a function can appear to have a largest and smallest value in its range and yet fail to have an absolute maximum or minimum because these values are not assumed for any points in the domain of the function. Let S = fx : ▯1 < x < 1g and f(x) = x . The graph of f looks as follows: 2 -1 1 It is clear that ▯1 < f(x) < 1 , but f never assumes the values ▯1 or +1 on S . Consequently f has no absolute minimum nor absolute maximum. We are ▯nished with examples for the moment but we proceed to explore the lesson of Example 8 with some care. If a real valued function f is to have an absolute maximum on a set S , there must exist a number M which is greater than or equal to all the values assumed by f and there must be a point p within S for which f(p) = M . For some purposes it is useful to have a de▯nition which embodies the ▯rst of the foregoing requirements. De▯nition. A function is bounded above on a set S if there exists a number M for which s 2 S implies f(s) ▯ M . Analogously, if a real valued function f is to have an absolute minimum on a set S , there must exist a number m which is less than or equal to all the values assumed by on points of S and a point p 2 S where f(p) = m . We are thus led to de▯ne: A function f is bounded below on S if there exists a number m for which s 2 S implies m ▯ f(s) . Example (8) above shows that a function can be bounded above and below yet have no absolute maximum or minimum. It would be useful to have concise conditions which guarantee the existence of an absolute maximum or minimum. Such conditions are known. They involve a condition to be satis▯ed by the function and a condition to be satis▯ed by the set. More precisely, the result is: Extreme Value Theorem: A continuous function on a closed bounded set has an absolute maximum and minimum. The extreme value theorem is covered in the text Haeussler and Paul in section 14.2. We shall add to that discussion. 3 Although the extreme value theorem is a very general result which applies to real valued functions with a domain in any number of dimensions, we shall be concerned with only the simplest instance of a real valued function on a subset of the real numbers. We are not able to rigorously prove the result because we do not have a su▯ciently complete description of the real numbers (in particular, we lack something called the least upper bound axiom). Instead we shall be satis▯ed with explaining what the result means and learning how to use it. We already have considerable experience and understanding of continuity since it has been covered in the main text for the course. The remaining terms in the hypothesis of the theorem are \closed bounded set". Meaning of BOUNDED A set of S real numbers is said to be bounded if there exists a number L which is less than all the numbers in S and another
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