mt1 - x,y,z and w . Use this inequality to prove that a 3 +...

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Homework 1, Math 245, Fall 2010 Due Wednesday, September 15 in class. The solution of each exercise should be at most one page long. If you can, try to write your solutions in LaTex. Each question is worth 2 points. 1. Let A,B,C be sets. Prove that A ( B \ C ) = ( A B ) \ ( A C ). 2. Let A = { x R : | x + 1 | ≤ 5 } . Show that A = [ - 6 , 4]. 3. Let f : R R ,f ( x ) = x 2 1+ x 4 . What is the image of f ? 4. Prove that 5 x 2 - 16 xy + 13 y 2 + 8 x - 12 y + 4 0 for any real numbers x and y . When does equality hold ? 5. Prove that x 4 + y 4 + z 4 + w 4 4 xyzw for any real numbers
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Unformatted text preview: x,y,z and w . Use this inequality to prove that a 3 + b 3 + c 3 3 abc for any real non-negative numbers a,b,c . (Hint: For the rst part, use the Arithmetic Mean-Geometric Mean inequality for two numbers. For the second part, make w := 3 xyz in the rst inequality.) 6. Bonus Question! Prove that: x 2 + y 2 + z 2 xy + yz + zx. for any real numbers x,y and z , When does equality hold ?...
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This note was uploaded on 12/07/2011 for the course MATH 245 taught by Professor Cioaba during the Fall '10 term at University of Delaware.

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