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Unformatted text preview: Brandy Hicks August 31, 2007 MTHSC 974.001  Homework # 2 1. For r > o , draw the graph of f 1 r ( X ) = 1 2 rx 2 ,  x  1 r  x   1 2 r ,  x  1 r to see if f 1 r is convex on R . 1.JPG Although the drawing is rudimentary, it appears that f 1 r ( x ) is convex on R (a) What happens as r + ? Proof: As r , f 1 r ( x )  . As r gets very large, 1 r gets very close to zero. Since the value of f 1 r ( x ) = 1 2 rx 2 when  x  1 r , then when the absolute value of x is between 0 and 1 r , which is very close to 0, we get 1 2 r (which remember is approaching ) x 2 (and x is very close to 0). This means that the equation basically becomes 1 2 1 10 100000000000 . Therefore this whole piece of the function is going to 0. Now we look at the the other segment of the function. We see that when  x  1 r ,  x   1 2 r . So  x  has to be greater than 1 r which is basically 1 , which is very close to zero....
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This note was uploaded on 10/24/2009 for the course MTHSC 974 taught by Professor Peterson during the Fall '07 term at Clemson.
 Fall '07
 Peterson

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