lect116_16rev_f11

lect116_16rev_f11 - Thursday October 13 Lecture 16 Maximum...

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Thursday, October 13 Lecture 16 : Maximum and minimum values of a function. (Refers to Section 3.4 and 4.4 in your text) Students who have mastered the content of this lecture know about : Local maxima and minima of a function f, absolute maxima and minima of a function f, extreme values of f, the EVT, differentiable functions being continuous, critical points, Fermat’s theorem. Students who have practiced the techniques presented in this lecture will be able to : Find the local maxima and minima of a function by first locating critical points, find the absolute maximum and minimum of a function by first locating critical points, 16.1 Definition Let f be a function with domain D . We say that f has an absolute (global) maximum at the number c in D if f ( x ) f ( c ) for all x in D . We say that f has an absolute (global) minimum at m in D if f ( m ) f ( x ) for all x in D . In this case we say that f ( m ) and f ( c ) are extreme values of f on its domain D . 16.2 Definition Let f be a function with domain D . Then we say that f has a local maximum at the number c in D if f ( x ) f ( c ) for all x in an open interval centered at c which intersects D . We say that f has a local minimum at the number m in D if f ( x ) f ( m ) for all x in an open interval centered at m which intersects D . - Note that a local maximum or minimum may be an “ endpoint ”. - It may also be on a part of the curve where there is not derivative: consider the curve of f ( x ) = | x | at x = 0 for example. 16.3 Example – Consider the function with domain D = [ 1, 4] whose graph is plotted below.
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With this graph the best we can do is approximate the absolute maximum and minimum. - We estimate that f ( x ) has an absolute maximum at x = 1 where the point ( x , f ( x ) ) = ( 1, 37) is the absolute maximum - We estimate that f ( x ) has an absolute minimum at x = 3 where the point ( x , f ( x ) ) = (3, 27) is the absolute minimum. -
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lect116_16rev_f11 - Thursday October 13 Lecture 16 Maximum...

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