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# solutions5 - CM221A ANALYSIS I Solutions to Exercise Sheet...

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CM221A ANALYSIS I Solutions to Exercise Sheet 5 Calculate the derivatives of each of the following two functions. Find all of the local and global maximum and minimum values, if there are any, of the functions on the intervals given. Sketch the graphs of the functions in these regions and explain your results! 1. f ( x ) = 1 1+log( x ) 2 on (0 , e]; f ( x ) is positive and vanishes as x + and also as x 0 + 0 . It is not defined at 0 and so has a g.l.b. but not a min. Its only turning point is where f 0 ( x ) = 0 , i.e. at x = 1 , which is its maximum. It has a local min at x = e , which is an end point of the interval. 2. f ( x ) = xe - x 2 / 2 on [0 , ); It is positive everywhere on the interval except at x = 0 , where it vanishes. Thus it has a local and global minimum at x = 0 . It converges to zero as x + . It has a maximum where its derivative vanishes, at x = 1 . 3. Evaluate the following derivative (you must state the conditions under which the result holds) d d x f ( x ) 2 g ( x ) 3 .

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solutions5 - CM221A ANALYSIS I Solutions to Exercise Sheet...

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