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Unformatted text preview: 15. Extreme Values of Functions P. K. Lamm 10/20/11 (16:24) p. 1 / 5 Lecture Notes: 15. Extreme Values of Functions These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. Extreme values of a function Let y = f ( x ) for all x in some set S . Definition: The absolute maximum or max value of f on S is the value M such that • f ( c ) = M , for some c ∈ S , and • f ( x ) ≤ M , for all x ∈ S . Definition: The absolute minimum or min value of f on S is the value m such that • f ( d ) = m , for some d ∈ S , and • f ( x ) ≥ m , for all x ∈ S . Definition: The extreme values or extrema of f on S are the absolute max and absolute min values of f on S . A natural question is the following. Is it the case that a given function f must always have a max/min on a set S? The answer is no, as Examples 1.1 and 1.2 below show: Example 1.1: Let f ( x ) = 1 x for x ∈ S = (0 , 1]. There is a min of 1 at x = 1 but no max value is attained. Note: A similar phenomenon can can occur even if f does not approach infinity as x → + . For example, if f ( x ) = 2- x , for x ∈ (0 , 1], then f has an absolute min value of 1 at x = 1; although f approaches 2 as x → + , f ( x ) never attains the value 2 for any x ∈ (0 , 1], so this particular function does not have an absolute max on (0 , 1]. Example 1.2: Let f be given by the graph to the right, with S = [0 , 1]. Again there is a min value of 1, this time occurring at x = 0 and x = 1. But no max value is attained. 1 15. Extreme Values of Functions P. K. Lamm 10/20/11 (16:24) p. 2 / 5 The type of behavior seen in the last two examples is ruled out if we restrict ourselves to a continuous function on a closed interval: Theorem: If f is continuous on a closed interval [ a,b...
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.
- Fall '10