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bmhicks_homework_2

# bmhicks_homework_2 - Brandy Hicks MTHSC 974.001 Homework 2...

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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. This then makes our function | x | (which is greater than 0) - 1 2 ·∞ .

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