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Unformatted text preview: 1 Maximum and Minimum Values This is all about optimization problems. Definition 1. A function f has an absolute maximum at c if f ( c ) ≥ f ( x ) for x in the domain of f . f ( c ) is called the maximum value. A function has an absolute minimum at c if f ( c ) ≤ f ( x ) for all x in the domain of f . f ( c ) is called the minimum value. Maximum and minimum values are called extreme values. *Draw an example The problem is that not all functions have absolute minima or maxima. However, if we zoom in, a function might have a value that’s larger than those around it. *Draw an example Definition 2. A function f has a local maximum at c if f ( c ) ≥ f ( x ) when x is near c . f has a local minimum if f ( c ) ≤ f ( x ) when x is near c . x is near c if there’s some interval around c where the statement is true. Example 1.1. f ( x ) = cos x f ( x ) = x 2 f ( x ) = 3 x 4- 16 x 3 + 18 x 2 How can we tell if a function has an extreme value or not? With a theorem, of course. Extreme Value Theorem. If f is continuous on a closed interval [ a, b ], then f attains an absolute maximum value f ( c ) and an absolute minimum f ( d ) for some values c and d in [ a, b ]....
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- Spring '08