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14209an5b.4-7 - c Kendra Kilmer Section 5.5 Absolute Maxima...

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c circlecopyrt Kendra Kilmer March 26, 2009 Section 5.5 - Absolute Maxima and Minima Definition: An absolute maximum is the largest value a function obtains on its domain. An absolute minimum is the smallest value a function obtains on its domain. Example 1: Find the absolute maximum and absolute minimum of each function if they exist. a) b) c) 4
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c circlecopyrt Kendra Kilmer March 26, 2009 Extreme Value Theorem: A function f that is continuous on a closed interval [ a , b ] has both an absolute maximum value and an absolute minimum value on that interval. Example 2: Find the absolute max and min for the functions in Example 1 on the interval [ 1 , 1 ] Finding Absolute Extrema on a Closed Interval 1. Check to make certain that f is continuous over [ a , b ] 2. Find the critical values in the interval ( a , b ) . 3. Evaluate f at the endpoints a and b and at the critical values found in step 2. 4. The largest value obtained from the previous step is the absolute maximum of f ( x ) on [ a , b ] and the smallest value obtained is the absolute minimum of f ( x ) on [ a , b ] . Example 3: Find the absolute extrema of the following functions on the given intervals a) f ( x ) = 2 x 3 3 x 2 12 x + 24 on [ 1 , 4 ] 5
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