{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

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 ﬁrst part, use the Arithmetic Mean-Geometric Mean inequality for two numbers. For the second part, make w := 3 √ xyz in the ﬁrst 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 ?...
View Full Document

{[ snackBarMessage ]}