20-max-min

20-max-min - Math 1a Maxima and Minima Fall, 2009 For each...

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Unformatted text preview: Math 1a Maxima and Minima Fall, 2009 For each of the following functions, find the absolute maximum or minimum on the given closed interval [ a,b ] by following these steps: 1. Find the critical numbers of f (that is, the numbers c so that f ( c ) is zero or undefined in the open interval ( a,b ). 2. Find the value of f at each of the critical numbers. 3. Find the value of f at each endpoint of the interval. 4. The largest value from the previous two steps is the absolute maximum of the function and the smallest is the absolute minimum . In each case, a simple sketch might help visualize the problem. For your convenience, Ive drawn each graph on the next page. 1 f ( x ) = 12 x- x 3 on [- 3 , 5] 2 f ( x ) = x 3- 6 x 2 + 9 x + 2 on [- 5 , 0] 3 f ( x ) = x 6- 36 x 4 on [0 , 7] 4 f ( x ) = ( x 2- 4 ) 2 / 3 on [- 3 , 2 . 5] 5 f ( x ) = 2 x 2 x 2- 4 on [- 1 , 1] 6 f ( x ) = sin( x ) on [- 2 , 3 ] Sometimes things are slightly more complicated. Either the interval isnt finite (or perhaps isnt closed) or the function isnt continuous on the entire interval. See if you can cope with the following. 7 f ( x ) = x 2- 5 on [- 2 , 4) 8 f ( x ) =- 4 + 2 x on [- 1 , ) 9 f ( x ) = x + 1 x = x 2 + 1 x on (0 , ) 10 f ( x ) = x 2 + 1 x = x 3 + 1 x on (0 , )- 5 5 ........................................................................................................ . . . . . . . . . . . .- 60- 40- 20 20 40 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................... . . . . . . . . . . . . . ................... . . . . . . . . . . . . . . . . ....
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.

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20-max-min - Math 1a Maxima and Minima Fall, 2009 For each...

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