For all real numbers x and y, $\displaystyle sqrt(xy) $ is less than or equal to

X + Y

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2

provide a counter example for statements that are false and provide a complete proof for those that are true

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- Jan 30th 2009, 06:29 AMtreethetaMathematical proofs
For all real numbers x and y, $\displaystyle sqrt(xy) $ is less than or equal to

X + Y

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2

provide a counter example for statements that are false and provide a complete proof for those that are true - Jan 30th 2009, 06:52 AMkalagota
- Jan 30th 2009, 06:54 AMo_O
Start with this: $\displaystyle (x-y)^2 \geq 0 \qquad \forall x, y \in \mathbb{R}^{\color{red}+}$

Playing around with it gives us:

$\displaystyle \begin{aligned}x^2 - 2xy + y^2 & \geq 0 \\ x^2 + 2xy + y^2 & \geq 4xy \\ (x+y)^2 & \geq 4xy \\ & \ \ \vdots\end{aligned}$ - Jan 30th 2009, 06:56 AMtreetheta
oh wow thanks

I was thinking so hard on that one too,

can you help me out with something else too i got a test in 2 hours xD

If we are asked to write the contrapostive of a conditional statement ( For each integer n, if $\displaystyle n^2$ is an odd integer, then n is an odd integer)

would the contra positive be replacing the odd's with even? - Jan 30th 2009, 07:00 AMChop Suey
- Jan 30th 2009, 07:08 AMtreetheta
so the contrapositive would be if n is even then n^2 is even?