MAC2311, Calculus I 4.1 Maximum and Minimum Values Some of the most important applications of differential calculus are optimization problems – finding the maximum and minimum values of functions. Here are some examples: 1. What is the shape of a can that minimizes manufacturing costs? 2. What is the maximum acceleration of a space shuttle? 3. What is the radius of a contracted windpipe that expels air most rapidly during a cough? 4. At what angle should blood vessels branch so as to minimize the energy expended by the heart in pumping blood? Definition: A function f has an absolute maximum (or global maximum) at c if ( ) ( ) f c f x ≥ for all x in D , where D is the domain of f . The number ( ) f c is called the maximum value of f on D . Similarly, a function f has an absolute minimum (or global minimum) at c if ( ) ( ) f c f x ≤ for all x in D , where D is the domain of f . The number
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